ARITHMETIC,

IN TWO PARTS.

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PART FIRST,

ADVANCED LESSONS IN MENTAL ARITHMETIC.

PART SECOND,

RULES AND EXAMPLES FOR PRACTICE IN

WRITTEN ARITHMETIC.

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FOR COMMON AND HIGH SCHOOLS.

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BY FREDERIC A. ADAMS,

PRINCIPAL OF DUMMER ACADEMY.

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Eighth Thousand

LOWELL:

PUBLISHED BY D. BIXBY & CO.

Boston: B. B. Massey & Co.: W. J. Reynolds & Co. New Your: D. Appleton & Co.

Philadelphia: Thomas, Cowperthwit & Co. Galtimore: Cushing & Brother.

Richmond: Nash & Woodhouse, Charleston: McCarter & Allen. Mo-

bile: J. Dobler. New Orleans: J. B. Steele. St. Louis: S. B. Meech.

Detroit: C. Morse. Chicago: A. H. & C. Burley. Provi-

dence: C. Burnet, Jr. Portland: Hyde, Lord & Duren.

1848

Entered according to Act of Congress, in the year 1846,

BY DANIEL BIXBY,

In the Clerk’s Office of the District Court of the District of Massachusetts.

STEREOTYPED AND PRINTED BY DICKINSON & CO., 52 WASHINGTON ST., BOSTON

PREFACE.

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The book here offered to Schools and Academies, had its origin in the urgent want the author has found, in the case of his own pupils, of a higher work on Mental Arithmetic. Such a work, he as thought, should be constructed with reference to several important objects.

It should habituate the pupil to perform, with ease and readiness, mental operations upon somewhat large numbers. \It should present these operations in their natural form, freed from the inverted and mechanical methods which belong of necessity to operations in written Arithmetic.

It should train the student to such a power in apprehending the relations of numbers, as shall give him an insight into the grounds f the rules of Arithmetic; and, consequently, shall release him from dependence on those rules; and should free him from the liability to those wide mistakes often made in written Arithmetic, which appear so absurd, and are yet too frequently to excite the teacher’s surprise.

A higher training in Mental Arithmetic would also, it is believed, prepare the members of our schools, when they should leave their studies and engage in the active pursuits of life, to solve mentally, and with ease and delight, al large share of those questions of business or curiosity, for which a process of ciphering is ordinarily thought indispensable.

The study of Arithmetic in the schools of this country received its best impulse, unquestionably, in the publication of “Colburn’s First Lessons.” So completely has this little book performed the work within its prescribed sphere, that there is little reason to desire a change in that particular, or to expect that the work will, for the present, be superseded. Whoever would now write a book of First Lessons in Arithmetic, must, it is believed, if he would write a good one, walk most of his way in the steps of one, at least, who has gone before him.

The “Advanced Lessons” are designed to continue and extend the course of discipline in numbers, which is begun in the elementary book above named. Consequently it requires, for its successful study, an acquaintance with the elements, as taught in that work, or in some other occupying essentially the same ground.

In all the mental calculations in large sums, it will be found a uniform characteristic of this work to begin with the highest order of numbers in the sum, -- hundreds before tens, tens before units. In this way, the numbers are presented in the same order in which they are presented in the common usage of our language. In most of the operations of written Arithmetic, however, the smallest number is taken firs; and thus a method is pursed, the reverse of what the genius of our language would naturally suggest. Another advantage of taking the highest numbers first, in Mental Arithmetic is, that we thus obtain a large approximation to the final answer, at the first step. When the first step, however, as in written addition, or multiplication, furnishes only the units of the answer, leaving the hundreds or thousands still unknown, only a minute fraction of the answer is at first obtained. It is too plain to require proof, that that method will be most interesting and gratifying to the mind, which secures the largest portion of the answer at the first step. Another advantage of the method here used is finding the fact, that we naturally make the higher order the standard, and the lower order takes its value in the mind from a comparison with the higher, as a certain part of it. Thus 150 is apprehended by the mind, as one hundred and half a hundred. This is not, indeed, the method of acquiring the idea of large numbers, but the method of acquiring the idea of large numbers, but the method of combining them after the idea has been acquired; consequently, it is the legitimate method of instruction, just as soon as the pupil is qualified to enter on the study of such combinations. If, now, we obtain the number of the highest order firs, we have a standard, under which all the succeeding orders naturally fall, and from a comparison with which they successively take their value. If we begin with units, however, and work upward through the higher orders, we obtain no standard; we must hold the successive numbers in suspense, until the last term shall furnish the nucleus for the group, -- the standard under which tall the lower orders shall take their rank.

It is on the basis of these facts, which are only indications of the laws of the mind, that, throughout the Mental part of this Arithmetic, the author has in all operations, taken the highest order of numbers first. The increased interest which the persevering use of this method will awaken in the minds of pupils, will be, to teachers, a better commendation of its correctness, than any more extended mental analysis.

There are other features of the Advanced Lessons which are, perhaps, sufficiently distinctive to justify their mention here; but as the truest test of a school book is its use in the school room, the work is referred to that ordeal.

The Second Part contains examples in Written Arithmetic on all the most important rules. They are designed to be sufficiently numerous to lead the student to ready and accurate practice in ciphering. In this Part the author has aimed to interest the scholar by furnishing him with natural and reasonable questions, and to aid both teacher and scholar by arranging them progressively.

The rules and explanations will, probably, be found sufficient, after a thorough mastery of the First Part. It is not necessary that the pupil completes the First Part before beginning the Second. He may carry on both Parts at the same time; but, under each particular head, the mental part should be thoroughly mastered before the written examples are begun.

The answers to the questions in the Second Part are given in a separate work. This course has seemed to the author, on the whole, the best, notwithstanding some incidental disadvantages that may arise from it. It will enable the teacher to oversee a much larger amount of work in Arithmetic, than he could otherwise attend to.

The Key will be bound up with the Arithmetic, for the use of teachers; and such copies will be lettered Teacher’s Copy.

The present contains a considerable number of examples more that the Third Edition, but no change in the numbering of the sections or of the examples, to occasion inconvenience to the teacher.

To aid in awakening a higher interest and zeal in this branch of study, the author will offer a few suggestions.

Let the key be used as little as the teacher’s necessities will permit.

Let original questions be proposed by the teacher in connection with every Section.

Each member of the class should be encouraged to propose original questions to be solved by the lass.

It will often be useful, especially in a review, to alter some one figure in the conditions of each question. This often produces a happy excitement, and gives quite an new zest to the study

Dummer Academy, April 18, 1846.

C O N T E N T S.

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PART FIRST.

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PART SECOND.

EXPLANATIONS

1. The sign = indicates equality; as 7 time 3=21.

2. The sing + indicates addition; as 15+7=22.

3. The sign - placed between two numbers, indicates that the latter number is to be taken from the former; as 9-4=5.

The larger number is called the minuend; the smaller, the subtrahend.

4. The sign x indicates multiplication; as 6x7=12.

The two numbers are called factors; the number multiplied is called the multiplicand; the number by which it is multiplied, the multiplier.

5. The sign  indicates that the number placed before it, is to be divided by the number after it; as 15  5=3.

The number to be divided is called the dividend; the number by which it is divided is called the divisor.

6. When a number is multiplied by itself, the product is called the second power of that number, or the square of it; as 2x2=4, which is the second power, or the square of 2; so 9 is the square of 3; 25 the square of 5.

7. When a number is multiplied by itself, so as to be taken 3 times as a factor, the product is called the 3d power, or the cube of the number; thus 8 is the cube of 2, for it is formed by multiplying 2x2x2; 27, or 3x3x3, is the cube or third power of 3; 125, or 5x5x5, is the third power of 5. The number thus used as a factor, is called the root of the power; thus 3 is the square root of 9, and the cube root of 27; 5 is the square root of 25.

The number of the power may be expressed by a small figure thus 2³ is the 3d power of 2; 3² is the 2d power of 3; 5³ is the 3d power of 5.

A straight line from the center to the circumference is called the radius.

The diameter is a line from side to side of the circle, through the center. It follows that the diameter is equal to twice the radius.

Any portion of the circumference considered by itself is called an arc.

A sector of a circle is a portion of it bounded by two radii and the arc between them.

A sphere is a solid bounded by a curved surface every part of which is equally distant from the center of the solid.

MENTAL ARITHMETIC

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PART FIRST.

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SECTION I.

MULTIPLICATION OF TENS AND UNITS.

1. A man drove six oxen to market, and sold three of them for 50 dollars apiece. What did they come to?

Three time 50 are 150. Ans. 150 dollars.

He sold the remaining three for 52 dollars apiece. What did they come to?

Three times 50 are 150, and three times 2 are 6, which added to 150 makes 156. Ans. 156 dollars.

What did they all come to?

Twice 100 is 200, and twice 50 is 100, which added to 200 makes 300, and 6 added to 300 makes 306. Ans. 306 dollars.

2. A merchant bought 45 barrels of flour for 6 dollars a barrel. What did it come to?

5 time 70 are 350; 5 times 5 are 25, which added to 350 makes 375. Ans. 375 dollars.

What did all the flour come to?

300 and 200 are 500, 70 and 70 are 140, which added to 500 makes 640, and 5 are 645. Ans. 645 dollars.

3 What will 87 barrels of flour come to at 6 dollars a barrel?

6 times 80 are 480, and 6 times 7 are 42, which added to 480 makes 522. Ans. 522 dollars.

4. What are 7 times 68? What are 8 times 72?

What are 9 times 84? What are 4 times 96?

8 times 64? 7 times 85? 5 times 79? 5 times 79?

4 times 98? 3 times 81? 6 times 73? 6 times 86?

The preceding examples will show the importance of being able readily to multiply tens by units. This becomes easy, after acquiring the Multiplication Table. It may be connected with a review of the Multiplication Table in the following manner.

Twice 1 are how many? Twice 10 are how many?

Twice 2 are how many? Twice 20 are how many?

A number which contains another number a certain number of times, is a multiple of that number.

Thus 6 is a multiple of 2; 15 of 3; 28 of 7.^*

Name all the multiples of 2, from 2 to 60.

Name the multiples of 20, from 20 to 600.

What are the multiples of 3 up to 75? of 30 up to 750?

What are the multiples of 4 up to 80? of 40 up to 800?

What are the multiples of 5 up to 100? of 50 up to 1000?

SECTION II.

MULTIPLICATION OF TENS AND UNITS. – COMPLEMENT.

1. What will 17 tons of hay come to at 8 dollars a ton? Ans. 8 times 10 are 80, and 8 times 7 are 56 added to 80 makes 136. 136 dollars.

2. What will 37 pounds of sugar come to at 9 cents a pound?

3. A man drove 87 sheep to market, and sold them for 6 dollars apiece. What did they come to?

4. A man traveled on foot 8 days; he traveled 29 miles each day. How many miles did he travel in all?

In each of the above examples the second product when added to the first makes a sum exceeding the next even hundred; thus, in the 1^st. example – 80+56; in the 2d. 270+63; in the 3d., 480+42; in the 4^th., 160+72.

In order to perform such examples with ease, quickness, and without mistake, each step in the process should be made the subject of distinct practice. to illustrate these steps by the first example, 80+56, the first thing to be done is to think of the number which must be added to 80 to make 100, namely, 20; the next is to take this 20 from 56, and what remains, --36, -- will belong to the next hundred.

The number which in such cases must be added to a given number to make up an even hundred may be called the Complement of that number. Thus the complement of 80 is 20; of 60, 40; of 90, 10; of 56, 44. What is the complement of 10? 30? 50? 70?

^* What is the complement of

How many are 40+76? 80+34? 70+91? 90+17? 25+83? 36+71? 45+82? 56+73? 43+82? 95+36? 37+84? 45+76? 88+37? 94+17? 76+_87?

To multiply any number less than 10 by 11, repeat the figure expressing the number; as 3 times 11 is 33, 4x11=44.

To multiply by 11 any number of two figures,. Think of the first figure, then of the sum of the two figures, then of the last figure. These three figures will express the answer.

Thus 11x23; the first, 2; the sum of the two, 5; the last, 3. Ans. 253. 11x24=264, 11x32=352; 11x43=475.

Remember, if the sum of the two is as much as 10, you must increase the first figure by one.

How many are 11x26? 11x28? 11x29? 11x41?

11x43? 11x45? 11x61? 11x62? 11x64? 11x71?

11x73? 11x81? 11x94? 11x75? 11x86? 11x89?

11x82? 11x84?

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SECTION III

PRACTICAL QUESTIONS.

1. If a rail-road car travels 23 miles in one hour, how far will it travel in 9 hours?

2. If a horse travels 38 miles in one day, how far will he travel in 6 days?

3. If a man earns 14 dollars a month, how much will he earn in 7 months?

4. If a man spends 6 cents a day for ardent spirit, how much will that amount to in 10 days? How much in 30 days? How much in 300 days? How much in 60 days? How much in 5 days? How much in 365 days?

5. If a man earns 10 cents in an hour, and works 12 hours in a day, how much will he earn in a week, there being 6 working days in a week? How much in 10 weeks? How much in 50 weeks?

6. If a scholar in school is idle 18 minutes in the forenoon, and 18 minutes in the afternoon, how much time will he lose in a week, if there are 6 forenoons, and 4 afternoons of school time in a week?

7. If a town is 6 miles long, and 5 miles broad, how many square miles does it contain? If there are 40 inhabitants on every square mile, how many inhabitants does the town contain? 40 times 30. 4 times 30 are 120. 40 times 30 are 10 times as many. If one in 12 of the inhabitants were able-bodied men, how many able-bodied men would there be? If one in 6 are able-bodied men, how many such are there?

8. What will 146 yards of broadcloth come to at 5 dollars a yard?

9. What will 86 yards of broadcloth come to at 6 dollars and a half a yard?

10. What will 740 barrels of flour come to at 6 dollars a barrel? at 5 dollars a barrel? at 5 dollars and a half a barrel?

11. What will 33 gallons of molasses come to at 31 cents a gallon? at 34 cents a gallon? at 40 cents a gallon?

12. What will 38 pounds of coffee come to at 14 cents a pound? at 16 cents a pound?

13. If a room is 14 feet long and 9 feet high, how many square feet are there in one of the side walls? How many in both the side walls? If the same room is 13 feet wide, how may square feet in one of its end walls? How many square feet in both its end walls. How many square feet in the ceiling?

14. A man wishes to know how many shingles he must buy in order to shingle his house. His house is 40 feet long, and it is 18 feet from the eaves to the ridge pole. How many square feet are there in one half of the roof? How many in the whole roof?

One thousand shingles will cover 10 feet square, how many thousand shingles will cover the roof? If shingles cost 4 dollars a thousand, how much must be paid for shingles enough to cover the roof? If the labor, the boards, and the nails, added together, cost as much as the shingles, what will be the whole expense of boarding and shingling the roof?

15. What are 8 ½ tons of hay worth, at 13 dollars a ton?

16. If one acre of ground produces 65 bushels of corn, how much would grow on 9 acres?

17. If an acre of ground produce 228 bushels of potatoes, how many bushels would grow on 5 acres?

18. If standing wood is worth 2 dollars a cord, what is the value of the wood on 7 acres, each of which furnishes 18 cords?

19. If there are 200 families in a town, and each family consumes 12 cords of wood annually, how may cords are used in the town each year?

What is the whole value of the wood at 3 ½ dollars a cord? How much money will be saved in the town if each family burns 2 cords less than before?

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SECTION IV

DIVISION

1. What is one half of 20? of 40? of 60? of 80? of 100? of 120? of 140? of 160?

2. What is one half of 22? of 42? of 62? of 82? of 102? of 112? of 142? of 162? of 182?

3. What is one half of 44? of 64? of 86? of 48? of 66? of 28? of 84? of 68? of 46? of 24? of 26? of 62?

4. What is one half of 70? divide it into 60 and 10.

What is one half of 90? divide it into 80 and 10.

What is one half of 50? of 30? of 110? of 150?

5. What is one half of 32? of 54? divide it into 50 and 4.

What is one half of 76? of 74? of 78? of 96? of 98? of 92? of 94? of 72? of 76? of 58? of 56?

6. What is one half of 43? One half of 40 is 20. One half of 3 is 1 1/2, this added to 20 makes 21 1/2.

What is one half of 47? of 49? of 63? of 65? of 67? of 69? of 83? of 85? of 87? of 89?

7. What is one half of 33? divide into 30 and 3.

What is one half of 35? 37? 39? 51? 53? 57? 59? of 71? of 73? of 75? of 77? of 79? of 91? of 93? of 95? of 97? of 99?

8. What is one half of 367? divide into 300, 60 and 7. What is one half of 674? of 895? of 724? of 632? of 945? of 424? of 688? of 546? of 392?

We can now find a very quick way of multiplying any number by 5. Take one half the number: multiply that by 10. We will take the numbers in question 2, and multiply them by 5 in this way.

Multiply 22 by 5: half of 22 is 11, and ten times 11 is 110.

Multiply 42 by 5: half of 42 is 21: ten times that is 210.

Multiply 62 by 5: half of 62 is 31: 310.

Multiply 82 by 5: half is 41: 410.

Multiply 102 by 5: half is 51: 510.

Multiply 112 by 5; half is 56: 560.

Multiply 122 by 5: half is 61: 610.

Multiply 142 by 5: half is 71: 710.

Multiply 162 by 5: half is 81: 810.

Multiply 182 by 5: half is 91: 910.

9. Multiply by 5 in this way the numbers in question 3. 44. 64. 86. 48. 66. 28. 84. 68. 46. 24. 26. 62.

You can, if you wish, perform these examples by both methods, and thus prove the work correct.

Multiply 862 by 5: half is 431: 4310. Multiply 672 by 5: half is 336: 3360.

10. Multiply 686 by 5. 748 by 5. 932 by 5. 896 by 5. 1262 by 5.

If the number to be multiplied is an odd number, so that half of it will show the fraction 1/2, this when you multiply by ten, will become 5: for ten halves are 5.

Multiply 781 by 5: half is 390 1/2: 3905 Ans.

11. Multiply 983 by 5: half is 481 1/2: 4815. Multiply 845 by 5. 381 by 5. 953 by 5. 845 by 5. 637 by 5. 429 by 5.

12. What is one fourth of 40? one fourth of 80? one fourth of 120? One fourth of 12 is 3, a fourth of 120 is 10 times as much, 30. What is one fourth of 160? one fourth of 200? of 240? of 280? of 320? 360? of 400?

13. What is one fourth of 60? Take half of it; then half of that half: half of 60 is 30, half of 30 is 15. What is one fourth of 100? one fourth of 140? one fourth of 180? one fourth of 220? one fourth of 260? one fourth of 300?

14. Another way of finding one fourth of the numbers in the last example , is as follows:

What is one forth of 60? Divide 60 into 40 and 20: one fourth of 40 is 10; one fourth of 20 is 5. 15.

What is one fourth of 100? divide into 80+20.

What is one fourth of 140? divide into 120+20.

What is one fourth of 180? divide into 160+20, &c.

15. What is one fourth of 30? of 50? of 70?

Find the best way of answering these, for yourself.

What is one fourth of 90? of 110? of 130? of 150? of 170? of 190? of 210? of 230? of 250?

16. What is one fourth of 76? divide the number into 40 and 36. What is one fourth of 96? divide the number into 80 and 16?

What is one fourth of 52? of 64? of 84?

17. What is one fourth of 368? There are several ways of dividing this number. First into 200+100+60+8; a second way would be, into 200+160+8; another way, into 320+48. This is shorter than either of the former. A better division still is into 360+8.

What is one fourth of 496? Into what different sets of numbers, each divisible by 4, can you divide this? What is one fourth of 964? of 336? of 836? of 596? 472? 1324? 1728? 2236?

18. What is one tenth of 10? of 20? of 30? of 40? of 50? of 60? of 70? of 80? of 90? of 100? of 110? of 120? of 130? of 140? of 150?

19. What is one tenth of 5? Ans. 5 tenths of one, or 5/10 equal to ½.

What then is one tenth of 15? of 25? 35? 45? 55? 65? 75? 85? 95? 14? 17? 36? 47? 52? 91? 43? 28? 65? 86? 47?

20. What is one fifth of 25? 40? 45? 50? 55? 60? 65? 70? divide 70 into 50 and 20. Of 75? divide into 50 and 25. Of 80? divide into 50 and 30. of 85? of 90? of 95? of 100?

21. What is one fifth of 64? of 82? of 91? of 67? of 73? of 59? of 63? of 72? of 78? of 83? of 87?

22. What is one fifth of 140? of 385? of 260? of 480? of 390? 580? of 470? 865? of 395?

23. The following is a short way of dividing a number by 5. Take one tenth of the number and double it. That of course gives 2 tents, which is equal to one fifth. Take the numbers in the last example, and divide by 5 in this way. One fifth of 140; one tenth is 14, double that is 28. One fifth of 385; one tenth is 38 and 5 tenths, twice that is 77. One fifth of 260; one tenth is 26, twice 26 is 52. One fifth of 480; one tenth is 48, twice that is 96. What is one fifth of 390? of 580? 470? 865? 395?

24. The following is a short way of multiplying a number by 25. Take one fourth of the number; multiply that by 100. This will give 100 fourths, which are equal to 25 whole ones.

Multiply 40 by 25; one fourth is 10, one hundred times that are 1000. Multiply 60 by 25; one fourth is 15, 1500.

Multiply 80 by 25; one fourth is 20, 2000.

Multiply 120 by 25; a fourth is 30, 3000.

Multiply 112 by 25; a fourth is 28, 2800.

Multiply 116 by 25, a fourth is 29, 2900.

25. Multiply 22 by 25; one fourth is 5 and a half; 100 times this are 5 hundred and half a hundred, 550.

Multiply 26 by 25; one fourth is 6½, 650.

Multiply 28 by 25; one fourth is 7, 700.

Multiply 30 by 25, 32 by 25; 34 by 25; 36 by 25; 40 by 25.

Multiply 42 by 25; 44 by 25; 46 by 25; 48 by 25; 50 by 25;

26. Multiply 13 by 25; one fourth is 3 and one fourth; one hundred times this is 300 and 3 fourths of a hundred or 75, 375.

Multiply 17 by 25; 19 by 25; 21 by 25; 23 by 25; 27 by 25; 29 by 25; 31 by 25; 33 by 25; 35 by 25.

27. Multiply 116 by 25; one fourth is 29, 2900.

Multiply 117 by 25; one fourth is 29 ¼, 2925.

Multiply 121 by 25; 87 by 25; 156 by 25; 960 by 25.

28.^* What is one third of 60? of 90? of 120? of 15? of 150? of 45? of 450?

What is one third of 18? of 180? of 21? of 210? of 36? of 360? of 30? of 390?

What is one third of 72? divide into 60 and 12.

What is one third of 54? of 85? of 98?

What is one sixth of 60? of 80? divide into 60 and 20. Of 74? of 84? of 96? of 100?

What is one sixth of 12? of 120? of 130? of 140? of 144?

What is one sixth of 18? of 180? of 200? of 210? of 220?

What is one sixth of 384? of 492? divide into 480 and 12. Of 555? divide into 540 and 15> Of 620? of 726? of 947?

What are the two factors of 18? of 180?

What are the two factors of 27? of 270? of 22? of 220? of 35? of 350? of 54 of 540? of 45? of 450? of 21? of 210? of 28? of 280? of 42? of 420?

30. What two numbers multiplied together will produce 24? What other two? What two factors will produce 240? What other two? What others?

What two factors will produce 30? What others?

What two factors will produce 300? What others?

What two factors will produce 18? What others?

What two factors will produce 180? What others?

Name all the pairs of factors that will produce 36? 360? 48? 480? 60? 600? 64? 640? 72? 720?

31. What is one ninth of 27? A man divided 270 dollars equally among 9 persons; how much did he give to each?

32. What is one fourth of 48? If 480 dollars are divided into 4 equal shares, what will each share be?

33. What is one seventh of 63? If a ship sails at a uniform rate, 630 miles in a week. how many miles does she sail in a day?

What is one ninth of 630? What is one sixth of 630? What is one third of 630?

34. What is one fifth of 25? If 250 trees are placed in 5 equal rows, how many will there be in each row?

35. What is one fourth of 36? In a circle there are 360 degrees; how many are there in one fourth of a circle? How many in one eighth of a circle? How many in one sixteenth of a circle?

36. What is one eighth of 56? If 560 trees are planted in 8 equal rows, how many would there be in each row? If planted in 16 rows, how many would there be in each row?

37. What is one eleventh of 55? If you place 550 trees in 11 equal rows, how many will the be in a row? If you place them in fifty rows, how many will there be in each row? If you place them in 25 rows, how many will there be in each row?

38. What is one twelfth of 96? If a man spends 960 dollars in a year, how much will be his average expense for each month?

39. What is one tenth of 40? 400? of 4000? of 80? of 800? of 8000? One fourth of 12? of 120? of 1200? of 12,000?

41. What is one fifth of 92? What is one third of 51? One fourth of 65? One fifth of 78? One sixth of 96? One seventh of 100? Divide into 70 and 30. What is one ninth of 117? Ans. One ninth of 90 is 10; one ninth of 27 is 3; 10 and 3 are 13.

What is one third of 49? one sixth of 84? one fifth of 79? one eighth of 100? one seventh of 91? one sixth of 79? one fourth of 76? of 92? of 60? of 52? of 65? of 70?

43. What is one fourth of 480? What is one fifth of 155? Divide into 150 and 5. What is one fourth of 920? What is one fifth of 15,765. This number may be divided into 15,000, 750, and 15, or 15,000, 500, 250, and 15.

44.^* What is one sixth of 4836? One eighth of 336? Divide into 320 and 16. What is one seventh of 574? one third of 684? one sixth of 43,248? one ninth of 72,108? of 64,827? one fifth of 5275? one fourth of 92,648?

45. What is one third of 6156? of 8436?

46. What is one fourth of 6428? of 9648?

47. What is one fifth of 7655? of 12,535?

48. What is one sixth of 13,218? of 1944?

49. What is one seventh of 10,542? of 14,280?

50. What is one eighth of 1632? of 2560?

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SECTION V.

TABLE OF TIME

60 Seconds, [sec.] make 1 Minute, marked m.

60 Minutes 1 Hour, h.

24 Hours 1 Day, d.

7 Days 1 Week, w.

4 Weeks 1 Month mo.

52 Weeks, 1 day, 6 hours 1 Year y.

365 Days, 6 Hours 1 Year y.

12 Calendar Months 1 Year y.

In common reckoning, 4 weeks are called a month, but this is merely for convenience in doing business. The number of days in a calendar month is 30 or 31; except February, which has 28 days, and in leap year 29. The 6 hours over and above the 365 days in a year, will in 4 years amount to a whole day; it is then added to February, making 29 days, and that year is called leap year. The number of days in may be seen in the line below. The months connected by a tie drawn over the words have 31 days; those connected by a tie underneath have 30.

┌─────────┐┌──────────┐┌────────┐┌────┐┌──────────┐┌─────────┐

Jan. Feb. March. April. May. June. July. August. Sept. Oct. Nov. Dec.

28.29. └──────────┘ └──────────┘

You observe that beginning with January, every alternate month has 31 days, till you come to July and August. Here there are two months together that have 31, and then the alternation goes on as before to the end of the year.

The leap year may be easily known from the fact that the number of they year is exactly divisible by 4.

What years in the present century have been leap years? What years will be leap years from now to the close of the century?

1. In one minute there are 60 seconds; in one hour there are 60 minutes. How many seconds are there in one hour? How many in 10 hours? How many in 20 hours? How many in one day? How many in seven days, or one week? How many in ten days? In 100 days? In 300 days? In 350 days? In 365 days?

2. If you save 30 minutes from idleness each day, how many hours will you save in a week? How many in 5 weeks? How many in 50 weeks? How many in 52 weeks?

3. If you read 40 pages each day, how many pages will you read in one week? How many in 10 weeks? How many in 52 weeks?

4. If a printer sets 4 pages of type in a day, in how many days will he set the type for a book of 500 pages? What will his wages come to at $1.50 a day?

5. If there are 300 members in the Legislature of Massachusetts, and each member receives 2 dollars a day during the session, what does the pay of all the members come to for one day? What does the pay of the Legislature amount to for one week? For 10 weeks?

6. The number of members in Congress is about 275. At 8 dollars a day, what is the amount of their pay each day? What would be the amount of their pay for 10 days? For 100 days?

7. How many days are there in the 3 months of spring? How many days in the 3 months of summer? How many days in autumn?

8. How many days in the winter of leap year? How many days were there in the winter of 1844? How many days in the winter of 1845?

9. If January comes in on Monday, on what day of the week will February come in?

If March comes in on Wednesday, on what day of the week will April come in?

If August comes in on Saturday, on what day of the week will September come in?

10. If April comes in on Sunday, on what day of the week will it go out?

If June comes in on Tuesday, on what day will it go out?

If September comes in on Saturday, on what day will it go our?

11. If January comes in on Friday, how many Sundays will there be in that month?

If it comes in on Thursday, how many Sundays will there be in that month? \If June comes in on Friday, how may Sundays will there be in that month? If it comes in on Saturday, how many?

If February comes in on Saturday, and that year is leap year, how may Saturdays will the be in the month? If it is not leap year, how many?

In 1845 February came in on Saturday how many Saturdays will there be in the month?

In 1844 February came in on Thursday, how many Tuesdays were there in that month?

12. If January comes in on Monday, on what day of the week will March come in, if it is leap year? On what day, if it is not leap year?

13. If June comes in on Wednesday, what day of the week will the 1^st of August be? The 9^th? the 12^th? the 15^th?

TABLE OF LINEAR MEASURE.

12 Inches, 1 foot, ft.

3 Feet, 1 yard, yd.

5 ½ yards, 16 ½ feet, 1 rod rd.

40 rods, 1 furlong, fur.

8 furlongs = 320 rods, 1 mile, m.

3 miles, 1 league, l.

69 ½, 1 degree of latitude, deg.

For lengths less than an inch, the inch is divided into fourths, eighths, tenths, or twelfths.

1. How many inches in 2 feet? In 4 feet? In 5 feet? In 7 feet? In 10 feet? In 12 feet? How many inches in 4 yards? In 1 rod? In 3 rods? How many feet in 1 furlong? In 2 furlongs? In 4 furlongs? In 1 mile?

2. How many miles in 46 leagues? In 132 leagues? How many miles in 2 degrees of latitude? In 3 degrees? In 4 ½ degrees? In 6 degrees?

In estimating the miles in any number of degrees of latitude, it is most convenient to call a degree 70 miles, and then, if we wish to be accurate, we may subtract from the answer half as any miles as there are degrees. In this way the distance of places from each other may be determined on a map; the degrees of latitude on the margin may be used as a scale of miles. If the distance of two places from each other is equal to 6 ½ degrees of latitude, how many miles are they apart?

3. How many yards in 10 rods? In 20 rods? In 30 rods? In 1 furlong? In 8 furlongs, or 1 mile?

4. In measuring land or a road with a chain 4 rods long, how may times must the chain be applied to the ground in measuring one mile? How many times in measuring the road from Boston to Salem, 15 miles? How many in measuring from Boston to Providence, 40 miles?

5. If a man walks three miles in an hour, how many minutes will he be in walking 1 mile? How many minutes in walking 1 fourth of a mile? How many rods will he walk in 1 minute? Ans. 16.

How many seconds will he be, then in walking 1 rod? 16 will go into 60, 3 times and 12 over. He will be, then , a little less than 4 seconds in walking 1 rod.

Let us now suppose he is precisely 4 seconds in walking 1 rod; how many rods would he walk in a minute? How many in 10 minutes? How many in 60 minutes? How many miles?

6. If a man in walking takes 6 steps to a rod, how many steps will he take in walking a mile? How many in walking 10 miles? How many in walking 40 miles?

7. If a man in walking takes 6 steps to a rod, and takes 2 steps in a second, how many seconds will he be in walking one rod? How many seconds in walking 10 rods? 20 rods? If a man walks 20 rods in one minute, how many minutes will it take him to walk a mile? 20 are contained in 320 just as many times as 2 are contained in 32.

8. If a man walks 20 rods in one minute, how long will it take him to walk 4 miles?

9. If a rail-road train goes 30 miles in an hour, how far does it go in one minute? How many rods in one second?

Ans. 30 miles in 60 minutes is 1 mile in 2 minutes; half a mile in one minute; quarter of a mile in half a minute; that is 80 rods in 30 seconds; that is 8 rods in 3 seconds; and in 1 second, one third of 8 rods or 2 rods and two thirds.

10. How many rods in 14 miles?

In one rod there are 16 ½ feet. In one mile there are 320 rods. how many feet are there in a mile? There are various ways of finding the answer to this question; some of them will be suggested, and the pupil left to take his choice.

First, how many feet are there in 300 rods?

This is not difficult, for in 3 rods there are three times 16 ½ feet, and in 300 rods there are 100 times as many. 3 times 15 feet are 45 feet; 3 times 1 ½ feet are 4 ½, which added to 45 make 49 ½ feet in 3 rods. Now 100 times 49 are 4900, and 100 halves are 50; 4950 feet in 300 rods. In 20 rods, there are 20 times as many feet as in 2 rods; in 2 rods, there are twice 16 ½ or 33 ; in 20 rods therefore there are 330 feet; 300 added to 4900 make 5200, and 30 added to 50 make 80; there are then 5280 feet in a mile.

Another method would be to multiply 320 first by 8, and that product by 2, for 8 and 2 are the factors of 16; then as there was ½ a foot in each rod left out, there must be added half as many feet as there are rods, or half of 320.

Another method would be to multiply 320 by 10, then by six and add the products, and lastly by ½ and add that to the other products.

The pupil can try each of these ways, and see if he obtains the same answer.

Let us now see if our answer is correct. If there are 5280 feet in a mile, how many are there in half a mile? One half of 5200 is 2600, one half of 80 is 40; there are then 2640 feet in half a mile. How many in 1 fourth of a mile? Now 1 fourth of a mile is 80 rods. If then there are 1320 feet in 80 rods how many will there be in 8 rods? One tenth as many. One tenth of 1320 is 132. How many are there in one rod? One eighth of 132; dividing 132 into 80 and 52; one eighth of 80 is ten, and one eighth of 52 is 6 4/8 or ½ , which added to 10 make 16½. We have now come down from 5280, and arrived by successive divisions to 16½, the number from which we started at first. The answer is thus proved to be correct.

11. How many feet are there in 2 rods? In 20 rods? In 200 rods?

12. How many feet are there in 3 rods? In 30 rods? In 300 rods? In 8 rods? In 80 rods, or a quarter of a mile?

13. How many rods are there in 2 miles? In 4 miles? In 8 miles? In 20 miles? In 30 miles? In 50 miles?

14. How many rods are there in half a mile? In three fourths of a mile? In one mile and a half? In one mile and three furlongs? In two miles and five furlongs? In 4 miles and 7 furlongs?

15. How many yards are there in 2 rods? In 20 rods? In 3 rods? In 30 rods? In 300 rods?

How many yards are there in 3 rods and 4 feet? How many yards are there in 17 rods 11 feet?

16. How many inches are there in 7 feet? In 9 feet? In 6 feet? In 8 feet and 6 inches? In 11 feet and 9 inches?

17. How many inches are there in 1 rod, or 16 ½ feet? How many inches in 2 rods? In 3 rods? In 4 rods?

18. A house is 46 feet and 5 inches in length, how many inches long is it?

A creeping vine grows on an average 3 inches a day; how many days will it take to grow from the grown to the top of a house that is 25 feet high?

19. A stage-horse travels 13 miles and 20 rods each day; how far will he travel in 60 days? How far in 120 days?

20. How far will the horse travel in a year, if he rests 5 days in the year?

21. What is the weight of iron used in one mile of railroad, allowing 55 pounds for a yard of rail?

One yard of heavy rail weighs 55 pounds. Twice this, or 110 pounds, would be the weight of both rails for one yard of a single track. Five and a half times this would be the weight for one rod. From this may be obtained the weight for 10 rods; for 100 rods; for 300 rods; for 320 rods.

22. What would be the cost of the iron for a single track of one mile of rail-road at 4 cents a pound? How much would be saved in the expense of the iron for one mile of rail-road, if the price of iron should be reduced one cent a pound?

23. If the cost of the iron for a single track of rail-road is 6000 dollars a mile, and the cost of the land and the labor of construction equals that of the iron, what would be the cost of 15 miles of rail-road? Of 24 miles?

24. What would be the cost of constructing one mile of common road at $2.25 a rod?

25. What would be the cost of building 80 rods of common wall at 54 cents a rod?

26. If a horse travels 10 miles in an hour, how long is he in traveling 1 mile? How long in traveling ¼ of a mile, or 80 rods? How many seconds is he in traveling 8 rods? How long in traveling 1 rod?

27. A body falling through the air, falls in the first second 16 ½ feet, and each succeeding second it falls twice 16 ½ feet further than in the preceding second. How far would a stone fall in 2 seconds?

28. How far would it fall in the third second? How far would it fall in 3 seconds?

29. How far would it fall in the fourth second? How fall would it fall in 4 seconds?

30. Sound moves through the air at the rate of 1090^ feet in a second. How many feet will it move in 3 seconds? How many feet in 4 seconds? How many feet in 5 seconds?

As sound is found thus to pass 5450 feet in 5 seconds, and as there are 5280 feet in a mile, we see that in 5 seconds sound moves 170 feet more than a mile. Now as 165 feet is just 10 rods, we say, without much error, that sound moves 1 mile and 10 rods in 5 seconds. This is accurate enough for all common purposes, and you will do well to fix it in your memory, and make your calculations from it.

31. How many rods will sound move in 1 second? One fifth of 320+10 rods = 66 rods.

32. How many rods in 2 seconds? How many rods in 4 seconds?

Thus, if you watch the stroke of an ax used by some one at a distance, and observe that the sound comes to you one second later than you see the stroke, you may know that the distance is 66 rods. If the sound of a bell comes to you two seconds after the stroke is given, you must be distant form the bell 132 rods. In these cases no allowance is made for the transmission of light. You are supposed to see the motion as soon as it occurs. This is not strictly the fact; but the time is so exceedingly small that it need not be taken into the account.

33. In a still night a church bell is sometimes heard at the distance of 12 miles; how many seconds, or nearly how many, after the stroke would the sound be heard at that distance:

34. If the report accompanying a flash of lightning is heard 4 seconds after the flash is seen, how far from the hearer was the discharge: How far, if the time between the flash and the report is 6 seconds; How far, if the time is 8 seconds: How far, if the time is 10 seconds? How far, if the time is 15 seconds?

35. The report of a cannon has, in some instances, been heard at the distance of 100 miles: in how many seconds, or nearly how many, after the discharge, would the report be heard at that distance? In how many minutes?

36. By means of a magnetic telegraph it is possible to communicate intelligence instantly from New Orleans to Boston, a distance of 1500 miles. If this intelligence could be communicated by sound passing through the air, how long would it be traveling that distance, allowing 5 seconds to the mile?

A ball discharged from a gun moves at first with a greater speed than sound, but it moves slower and slower, and before it is spent the report overtakes it, and passes by it: for sound moves always at the same rate.

37. If a cannon ball moves a mile in 8 seconds, how long would it be in moving 3 miles? How long in moving one fourth of a mile? How long in moving one eighth of a mile? How long in moving 1 ¾ miles?

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SECTION VI.

TABLE OF FEDERAL MONEY.

This is established by law as the currency of the United States.

The general mark for Federal Money is $, as $5.14, five dollars fourteen cents. A period must always be placed between dollars and cents.

1. How many mills in 2 cents? In 10 cents? In 12 cents? In 5 ½ cents? In 12 ½ cents? In 36 cents? In 1 dollar?

2. How many cents in 5 dimes? In 11 dimes? In 16 dimes? In 4 ½ dollars? In 17 ¼ dollars? In 12 ¾ dollars?

3. How many dimes in 7 dollars? In 13 ½ dollars? In 3 eagles? In 56 dollars? In 100 dollars?

4. How many cents in 35 mills? In 180 mills? In 600 mills? How many dimes in 80 cents? In 210 cents? In 740 cents?

5. How many dollars in 350 cents? In 325 cents? In 700 cents? In 850 cents? In 1400 cents? In 1675 cents? In 925 cents?

TABLE OF STERLING MONEY.

This is the currency of Great Britain.

1. How many farthings are there in 3 pence? In 7 pence? In 8 pence? In 10 pence? In 11 pence?

2. How many pence in 2 shillings? In 12 shillings? In 15 shillings? In 18 shillings? In 16 shillings?

3. How many shillings in 4 pounds? In 7 pounds? In 18 pounds? In 36 pounds? In 84 pounds?

4. How many farthings in 1 shilling and 6 pence? In t shillings and 6 pence? In 15 shillings and 4 pence?

How many pence in 10 shillings? In 20 shillings? In 2 pounds? In 4 pounds? In 12 pounds?

5. How many farthings in 1 pound? In 5 pounds? In 8 pounds? In 1 pound 2 shillings?

6. How many pence in 45 farthings? In 128 farthings? In 464 farthings? In 1296 farthings? In 648 farthings?

7. How many shillings in 80 pence? In 67 pence? In 372 pence? In 649 pence? In 840 pence?

8. How many pounds in 267 shillings? IN 845 shillings? In 432 shillings? In 640 shillings? In 4000 shillings?

9. How many pounds in 890 pence? In 16,000 farthings? In 720 pence? In 1200 pence? In 456 pence?

10. How many pence in 5 pounds 4 shillings? In 7 pounds 8 shillings? In 12 pounds 3 shillings?

How many farthings in 4 shillings 6 pence? In 9 shillings? How may farthings in 6 pounds 3 shillings 8 pence?

11. A man set out on a journey with £4 8s 6d in his pocket: before spending any thing, he received in payment of a debt £2 3s 8d. How much had he then? When he arrived home he had spent £1 4s 6d. How much had he then. These denominations, you must bear in mind, have not the same value in English currency, that they have in the United States.

In our country they have different values in the different States, but in none of them so high a value as in England. In the New England States a shilling is equal to 16 cents and two thirds, and 6 shillings make a dollar. In New Your 12 and a half cents are a shilling, and 8 shillings a dollar. In other states the values are still different; but these denominations are gradually giving way to those of the Federal Currency. They are now used only in naming prices. Accounts are not kept in them, and all that is important in them may be learned by practice without further notice here.

In the Sterling currency, used in England, a pound is equal to 4 dollars, 44 cents and 4 mills; 10 shillings, therefore or half a pound, are 222 cents, 2 mills. An English six pence is, therefore, 11 cents 1 mill. The following table will be useful in exchanging English money to our own.

1 pound, £, is $4.44 4

10 shillings, or half a pound, 2.22 2

1 shilling, .22 2

6d, or half a shilling, .11 1

4 shillings, 6 pence, 1.00 0

1 guinea, 21 shillings, 4.66 6

The actual value of the English money is a little higher than is here stated, but this is sufficiently accurate for a general table.

1. What is the value in dollars and cents of 2£? 3£? 4£? 5£? 1£ 6s? 2£ 8s? 3s 6d? 5s 9d?

TABLE OF DRY MEASURE.

2 pints, [pt.] make 1 quart, marked qt.

4 quarts 1 gallon gal.

8 quarts 1 peck pk.

4 pecks 1 bushel, bu.

8 bushels, 1 quarter qr.

36 bushels, 1 chaldron, ch.

These denominations are used for measuring grain, fruit, and coal. The pint, quart, and gallon are larger than the same denominations in wine measure, and less than those of beer measure.

1. How many pints in 1 peck? In 3 pecks? In 1 bushel? In 3 bushels? In 4 bushels?

2. How many quarts are there in 1 bushel? In 4 bushels? How many pecks in 7 quarters? In 2 chaldrons?

3. If a horse eat 4 quarts of oats each day, how many bushels will he eat in 10 weeks? How many bushels in 50 weeks? In 52 weeks?

What will they cost at 50 cents a bushel?

4. In 80 quarts how many pecks? How many bushels?

In 644 quarts how many pecks? How many bushels?

In 7840 quarts how many pecks? How many bushels?

5. In 100 pints how many pecks? How many bushels?

In 620 pints how many pecks? How many bushels?

TABLE OF AVOIRDUPOIS WEIGHT

These denominations are used in weighting hay, grain, meat, flower, ad all the most common articles bought ad sold by weight. On account of the waste in handling such articles, their shrinking in drying, and worthless admixtures sometimes found in them, 112 pounds are sometimes allowed for one hundred weight; this makes 28 pounds one quarter, and is called gross weight. In all the following questions of Avoirdupois wt., understand gross wt., unless net wt is expressed.

1. How many drams in 3 oz.? In 5 oz.? In 8 oz.? In 11 oz.? How may oz. In 12 pounds? In 15 lbs.? In 20 lbs.? In 32 lbs.? In 45 lbs.?

2. How many lbs. In 4 cwt. net weight? In 4 cwt. gross? In 6 cwt. net weight? In 6 cwt. gross? In 5 cwt. 2 grs. net? In 5 cwt. 2 grs. gross? In 7 cwt. 3 grs. net weight? In 7 cwt. 3 grs. gross?

3. How many lbs. in a ton, net weight? In a ton, gross? How many pounds in 5 tons, 3 cwt., net" In 5 tons, 3 cwt. gross?

4. There are 2 loads of hay whose net weight is as follows; the first, 25 cwt. 3 qrs. 17 lbs.; the second, 17 cwt. 2 qrs. 21 lbs. What is the weight of both?

5. A man set out for market with a load of hay weighting 36 cwt. 2 qrs. 15 lbs., net weight; he lost a part of it; the remainder weighed 25 cwt. 1 qr. 8 lbs. How much did he lose?

6. If there are 196 lbs. in a barrel of flour, how many pounds net weight are there in 10 barrels?

7. How many pounds are there in 100 oz.? In 650 oz.?

8. A barrel of flour weights 7 quarters gross; how many tons gross, are there in 100 barrels of flour?

9. What will be the expense of transporting by rail-road 100 barrels flour, 100 miles, at the rate of 3 dollars a ton?

What will be the expense of transporting a single barrel?

100 barrels are 700 qrs. gross weight, 400 qrs. = 100 cwt. = 5 tons: 300 qrs. = 75 cwt. = 3 tons, 15 cwt.; this added to 5 tons, makes 8 tons, 15 cwt.

10. The freight of goods by wagon is about 20 dollars a ton gross for 100 miles; at this rate what will be the cost of carrying a barrel of flour 100 miles?

TABLE OF TROY WEIGHT.

This is used for weighing gold and silver. The pound Troy is nearly one fifth less than the pound Avoirdupois.

1. How many grains in 6 pennyweights? In 8 pennyweights? In 12 pennyweights? In 1 oz.? In 2 oz.? In 4 oz.? In 6 oz.?

2. How many penny weights in 8 oz.? In 11 oz.? In 1 lb.? In 3 lbs.? In 8 lbs.? In 5 lbs.? In 1 lb. 3 oz.? In 2 lbs. 5 oz.?

3. How many oz. in 120 dwt.? In 480 dwt.? In 960 grs.? How many lbs. in 100 oz.? In 860 dwt.? In 1200 dwt.?

TABLE OF APOTHECARIES' WEIGHT

This table is used only by apothecaries in mixing medicines. The pound and ounce are the same as in Troy weight.

TABLE OF CLOTH MEASURE.

1. How many inches in 1 qr.? In 1 yd.? In 3 yds.? In 1 Ell. Eng.? In 1 Ell. Fr.? In 1 Ell. Fl.?

2. How many inches in 4 yds.? In 7 yds? In 12 yds? In 10 yds.? In 20 yds.? In 6 yds. 3 qrs.? In 4 yds. 1 qr.?

TABLE OF WINE MEASURE.

This table is used for measuring wine, spirits, cider, and water.

1. How many gills in 1 quart? In 1 gal.? In 4 gals.? In 6 gals.? In 10 gals.? In 13 gals.? In 15 gals.?

2. How many pints in 1 gal.? In 4 gals.? In 20 gals.? How many qts. in 1 barrel? In one hogshead?

3. How many gallons in 5 barrels? In 8 barrels? How many gals. in half a barrel? In one fourth of a barrel?

4. In 100 gals. how may barrels? In 300 gals. how many bls.?

5. At 14 cents a gallon, what is 1 qt. of vinegar worth? 3 qts.? 6 qts.? 10 qts.? 15 qts.? 21 qts.? 30 qts.?

6. What is one barrel of vinegar worth at 15 cts. a gallon? How much, if the price is 20 cts. a gal.?

A TABLE OF ALE OR BEER MEASURE.

(Used in measuring malt liquors, and milk.)

The beer gallon is a little more than one fifth larger than the wine gallon. There are other measures of beer besides those in the tables; as the firkin of 9 gallons; the kilderkin, 18; the hogshead, 54; but these are not much use in this country. A barrel of wine contains not quite three fourths as much as a barrel of beer.

1. In 1 bl. how many pints? How many pints in 3 bls.? How many gallons in 5 bls.? In 12 bls.? In 15 bls.? In 21 bls.?

2. In 100 gallons, how many bls.? How many bls. in 400 gals.? First consider how many bls. there are in 360 gals.?

MEASURE OF THE CIRCLE.

Every circle is supposed to have its circumference divided into 360 equal parts, called degrees; and each degree into 60 seconds. Whether the circle is great or small, it is still divided into 360 degrees; a degree therefore is always the same fixed part of the circumference of a circle, although its actual length is longer or shorter, according as the circle is great or small. The line passing from the center to the circumference is called the radius of the circle. To give you some idea of the length of a degree in circles of different magnitudes, I will state that, on comparing a degree in any circle with its radius, it has been found to be about one fifty-eighth part of it. In other words, 58 degrees on the circumference of a circle are about equal to the radius. If a degree is 1 inch, the radius of that circle is 58 inches. If the radius of a carriage wheel is 29 inches, a degree on the rim of the same wheel will be half an inch.

If we take for illustration one of the largest sized water wheels, 29 feet in diameter, a degree on its rim would measure only 3 inches.

You may enlarge the circle in your mind, till you suppose it extending over a plain, with a radius of 58 rods; a degree on such a circle will measure 1 rod. If the radius is 58 miles, a degree will measure 1 mile. Now the circle round the earth is so great that a degree measures 69 ½ miles. This may aid you in forming a conception of the vast magnitude of the earth.

Each of these degrees is divided into 60 minutes, or geographical miles; a geographical mile therefore is one sixth greater than a common mile. The table of circular measure is as follows:

The term miles instead of minutes, can be used only in reference to the great circle of the earth.

As the earth turns round on its axis once in 24 hours, every place upon it passes in that time through the 360 degrees of its circle; and on the equator, which is the great circle, each of these degrees, we have seen is 69 ½ miles.

How swiftly then does a body lying on the equator move in consequence of the daily revolution of the earth?

In 24 hours it passes through 360 degrees; in one hour then it will pass through one twenty-fourth part as many, which is 15 degrees. If it pass through 15 degrees in one hour, how many minutes will it be in passing through 1 degree? One fifteenth of 60 minutes is 4 minutes. If it pass through a degree in 4 minutes, what part of a degree will it pass through in 1 minute? One fourth of a degree, or 15 geographical miles. If it pass through 15 geographical miles in 1 minute, in how many seconds will it pass through 1 geographical mile? In 4 seconds; and in 1 second it will pass through one fourth of a geographical mile.

Now a geographical mile on the equator is, as we have seen, longer than a common mile. We will here suppose it no longer, but of the same length, and it appears that an object on the equator moves, as the vast earth whirls round on its axis, one quarter of a mile every second of time. Reflect now, that, while the surface of the earth moves with such amazing speed, so vast is its size, that it occupies an entire day and night in turning once round.

If, as above stated, the earth turns from west to east at the rate of fifteen degrees in an our, we can, by knowing the time of day in any place, ascertain what time it is at a place any particular number of degrees east or west of it. It is noon at any place when the meridian of that place passes under the sun.

1. When it is noon at Boston, what time is it at a place 15 degrees west of Boston? At a place 15 degrees east of Boston?

2. When it is 12 o'clock at Boston, what time is it at a place 1 degree west of Boston? At a place 1 degree east of Boston? At a place 2 degrees west of Boston? At a place 2 degrees east of Boston? 3 degrees east? 3 degrees west? 4 degrees east? 4 degrees west? 5 degrees east? 5 degrees west?

3. Indianapolis is 15 degrees west of Boston; when it is noon at Boston, what time is it at Indianapolis? When it is sunset at Boston, where will the sun be at Indianapolis?

4. Niagara Falls is 8 degrees west of Boston; when it is noon at Boston, what time is it at Niagara Falls? When it is 4 o'clock at Niagara Falls, what time is it at Boston?

5. Washington city is 6 degrees west of Boston; if you set your watch with the sun at Boston, and then carry it to Washington, your watch keeping accurate time all the while, when you arrive at Washington, will it be too fast or too slow? and how much?

6. Two travelers met at a public house; when one of them said to the other, "Friend, are you traveling east or west?" "I am direct from home," said the other, "where my watch agrees exactly with the sun, but here I find it is 10 minutes too fast: now if you can tell which way I am traveling you are welcome to know."

Had he traveled east, or west? and how far?

7. Boston is 71 degrees west of London; when it is noon at Boston, what time is it in London?

8. The English convicts are transported to Botany Bay, 150 degrees east of London; when it is noon at London, what time is it in Botany Bay?

9. English traders are settled on Columbia river, 120 degrees west from London; what time is it there when it is noon in London?

10. If a man is on the equator, which way must he travel, and how many geographical miles, to have the day 4 minutes longer than 24 hours? How far to have the day 2 minutes longer? How far to have it one minute longer? How far must he travel to have the day one minute and a half longer? Which way must he travel and how far, to have the day one minute shorter? 2 minutes shorter? 5 minutes shorter?

11. Suppose tow birds start from the same place on the equator, and fly, one east and the other west, at the rate of 60 geographical miles an hour, and at the end of the hour it is just sunset to the bird flying east; how high is the sun then at the place where the other bird is?

How high was the sun at the place of their starting, when they set out?

12. A shipmaster sails from New York for Europe, and for three days it is so cloudy that he cannot see the sun; on the fourth day he takes and observation of the sun at noon; and by his chronometer, which gives the New York time, it is half past eleven; how many degrees east from New York has he sailed?

In what longitude is he then, if New York is 74 1' west from Greenwich?

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SECTION VIII.

PRIME NUMBERS.

Numbers may be divided into two great classes. The first class comprises such numbers as cannot be formed by the multiplication of any two or more numbers together as 1, 2, 3, 5, 11, 17. These are called Prime numbers. The other class may be formed by multiplying 2 by 2; 6, which is equal to 2x3; 10 which is equal to 2x5, &c. These are called Composite numbers. These may always be formed by multiplying two or more prime numbers together. Thus all numbers are either prime, or are formed by the multiplication of Prime numbers together.

In separating numbers into their factors care should be taken that the factors be all prime. This in resolving 30 into its factors we may say it is formed by multiplying 5 by 6, but this is not sufficient, for 6 is not prime; it is formed of factors 2 and 3. The prime factors of 30 therefore are 2, 3 and 5. We may say that 30 is formed of the factors 3 and 10, but here again the analysis is not complete, for 10 is not prime; it is composed of the factors 2 and 5. Thus we are brought to the same three factors as before, namely 2, 3 and 5.

The following table of numbers from 1 to 100 will show what of them are prime, and what are the prime factors of those which are composite. This table should be carefully studied and made perfectly familiar. The analysis of composite numbers into their prime factors lies at the foundation of some of the most important operations in numbers, and affords an insight into some of the most intricate rules of Arithmetic.

On examining this table several things may be observed.

1. All the even numbers are composite; for they are all divisible by 2. So it appears in the table, with the exception of the number 2, which is regarded as prime because it is divisible only by itself.

2. Several of the numbers given above are powers of their prime factors. Thus 4 is the 2^nd power of 2, 8 the 3^rd power of 2, 16 the 4^th power of 2, 32 the 5^th, 64 the 6^th. 9, 27 and 81, are the 2^nd, 3^rd and 4^th powers of 3. 25 is the 2^nd power of 5, 49 the 2^nd power of 7.

3. If you double the number of times a factor is taken, you obtain the square of the number they at first made. Thus 4 is obtained by taking 2 twice as a factor. If you take it twice as many times, that is, 4 times as a factor, you obtain 16, which is the square of 4.

9 is obtained by taking 3 twice as a factor. If you double the number of times it is taken, thus 3x3x3x3, you obtain the square of 9.

8 is obtained by taking 2 three times as a factor. If you take it 6 times you obtain 64, the square of 8.

So universally, if you double the number, of times a factor is taken to produce a certain number, you obtain, not twice that number, but the square of it.

I will make a single remark here about the prime numbers, and then call your attention to the composite numbers.

Since the prime numbers are not formed by multiplying any two or more numbers together, they cannot be divided by any number. You will observe, however, that any number whatever may be divided by itself, and may also be divided by 1; but 1 is a unit and not a number; and by dividing a number by itself, or by 1, you obtain no new number. Dividing the number by itself you obtain 1, and dividing by 1, you obtain the number itself. Such an operation, therefore, brings out nothing new. It is only another way of expressing what was just as plain before. In the same way we may sometimes regard a number as produced by multiplying itself into 1; thus 7=7x1; but this is not multiplication, but only an expression in the form of multiplication. It produces no new number, and is employed only for convenience in order to make the reasoning more plain.

Composite numbers can be divided by their factors. Thus you can divide 10 either by 2 or by 5, and by no other number. If you divide by 2, you obtain 4 for the answer, or quotient; if you divide by 5 you obtain 2 for the answer. Dividing by a number then, is the same as erasing that number as a factor, and will always give for the answer the other factor, or factors. Thus dividing 10 by 2 you may represent thus, 2x5, leaving the factor 5 for the answer: dividing 10 by 5, thus, 2x5, leaving 2 for the answer. Divide 21 by 3, thus, 3x7. Divide 12 by 3 thus, 4x3 or 2x2x3.

It is plain, therefore, that if you express any number by its factors you can at once see what numbers you can divide it by. You can divide it by each of its prime factors, or by any combination of them, and by no other number. Thus 6=2x3, you can divide by 2 or by 3; 8=2x2x2, you can divide by 2, and that quotient by 2, and that by 2 again; 30=2x3x5, you can divide by 2, or 3, or 5, or by any tow of them combined.

Any composite numbers may be divided by any of its prime factors, or by any combination of them.

By what numbers can you divide 15? 18? 20? 21? 26? 27? 36? 42? 46? 48? 49? 50?

Sometimes we have two numbers, and we wish to know if there is any number that will divide them both. This we can ascertain if we express each number by means of its prime factors, and then see if the same factor is found in both; if so, they are both divisible by that number. Thus, if we wish to know whether any number will divide both 9 and 15, we express them thus, 3x3 and 3x5. Now 3 appears as a factor in both; they can both therefore be divided by 3. This number 3 is called the common divisor, because it is a divisor common to several numbers. If we wish to know whether any number will divide both 15 and 8, we express 15 by its factors, 5x3; and 8 by its factors 2x2x2. Now there is no factor common to both; no number therefore will divide them both; in other words they have no common divisor. Numbers which have no common divisor are said to be prime to each other. They may be composite consider by themselves, as in the case with 8 and 15, but if they have no common divisor they are said to be prime to each other. Numbers which have a common divisor are said to be composite to each other. If there are more than two numbers they must be treated in the same way. Each must be written in the form of its prime factors, and then, if any one number appears as a factor in them all, they are divisible by that.

Is there any common divisor to 9, 14 and 27? Written in the form of their factors they stand thus, 3x3, 2x7, 3x3x3. They have therefore no common divisor; for, though 3 or 9 will divide both the first and third number, it will not divide the second; and neither 2 nor 7, which are the factors of the second number, appear in the first or third. 2, 14, and 27 are, therefore prime to each other.

What is the common divisor of 15 and 27? of 14 and 22? Of 21 and 49? Of 35 and 28? Of 6 and 21?

Let us now take the following question: What is the common divisor of 18 and 30? By inspecting their factors 2x3x3, and 2x3x5, we find that 2x3 or 6, is common to both; 6 is therefore the greatest common divisor.

What is the greatest common divisor of 18 and 27? Of 4, 8 and 36? Of 15 and 45? Of 27 and 45? Of 40, 64, and 16? Of 44 and 24? Of 75 and 15? Of 80 and 100? Of 60 and 24? Of 35, 21 and 49? Of 15 and 50?

We have seen that a composite number can be divided only by its factors; and that prime numbers cannot be divided at all. It is frequently necessary, however, to attempt the division of prime numbers; and to divide composite numbers by some number different from their factors. For example we may wish to divide 9 by 4, or to obtain one fourth of 9. Now 4 is not a factor of 9, and the actual division of 9 by 4 is strictly speaking impossible. We proceed in this way. We divide 8 by 4, and obtain 2 for the answer, and we have a remainder of 1 which we have not divided. To show that we design this to be divided by 4 we write the 4 under it with a line between, thus ¼. In this way we indicate plainly enough what the answer is, although we have no one figure that will express it. This operation introduces us to a new class of quantities called Fractions. Fractions are expressions for quantities less than a unit. The word Fractions here means the same as broken numbers. In this class of expressions each unit is regarded as broken up, or divided into a number of parts. The figure below the line shows into how many parts the unit is divided; the figure above the line shows how many of those parts are taken; (or, what is just equivalent, the number above the line is regarded as divided by the number below it.) The fraction 3/7 indicates that each of the 3 units is regarded as divided into 7 equal parts; and that one of these parts is taken from each of them. The number below the line is called the Denominator; that above the line, the Numerator. If the Numerator is just equal to the Denominator, as 2/2, 5/5 7/7, the value of the fraction is just equal to 1. If the Numerator is smaller than the Denominator, the value of the fraction is less than 1, and is called a proper fraction; if the Numerator is greater than the Denominator, the value of the fraction is greater than 1, and is called an improper fraction. This, however, many always be changed to a whole number, or whole number and a proper fraction. Hence the propriety of the definition, that fractions are expressions less than unity.

Questions.

What is meant by a Common Divisor?

What is meant by the greatest Common Divisor?

When are numbers prime to each other?

When are number’s composite to each other?

What is the process of dividing 13 by 4?

In dividing 16 by 5? In dividing 25 by 6?

What are fractions?

Explain what is signified by each of the numbers in the fractions 2/3, In 4/7. In 6/11. In ¾. In 15/16.

A man bought a barrel of flour, and gave away two fifths of it; what fraction will express what he gave away? What fraction will express what he kept?

A man bought a load of hay, and sold two elevenths of it; what fraction will express what he sold? What fraction will express what he kept?

What is a proper fraction? Give an example.

What is an improper fraction? Give an example.

When is the value of a fraction just equal to 1?

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SECTION VIII.

MULTIPLICATION AND DIVISION OF FRACTIONS

We have seen that a fraction is not a simple expression, but composed of two numbers; and its value cannot be determined by one of these numbers alone, but by both taken in conjunction. By looking at the numerator, you cannot tell the value of the fraction unless you know what the denominator is. By looking at the denominator you cannot tell the value of the fraction, unless you know what the numerator is.

Let us now observe the effect of altering one of the terms of the fraction without altering the other. We will take the fraction 2/5. If we increase the numerator by 1, making it 3/5, we increase the value of the fraction, for we take one fifth more than we had before. So, if we multiply the numerator by 2, making it 4/5, we double the value of the fraction; and so of any other numbers, if we multiply the numerator, we multiply the value of the fraction. And, by the same reasoning, if we divide the numerator by 2, we divide the fraction by 2, for 1/5 is plainly one half as great as 2/5. So of all other numbers, by dividing the numerator we divide the fraction.

Let us now observe the effect of altering the denominator. If we increase the denominator of the fraction 2/5 by 1, making it 2/6, we have not increased the fraction, but diminished it, for one sixth is less than one fifth, and any number of sixths are less than the same number of fifths. We will multiply the denominator of the fraction 2/5 by 2 making it 2/10. What has been produced on the value of the fraction? One tenth is half as great as one fifth; and two tenths are half as great as two fifths. The fraction is therefore half as great as it was before; that is, it has been divided by 2. Multiplying the denominator, therefore, divides the value of the fraction.

We will now divide the denominator. Take the fraction 3/8; dividing the denominator by 2, we have ¾. Now as this is twice as great as 3/8, we have multiplied the fraction, by dividing the denominator.

There are, then, two ways of multiplying a fraction. We may multiply the numerator; or, if the multiplier is a factor of the denominator, we may divide the denominator. Thus, to multiply 3/8 by 2, we may multiply the numerator, which gives 6/8, or divide the denominator, which gives ¾, equal to 6/8.

To divide a fraction, we may either divide the numerator, if the divisor is a factor of it; or we may multiply the denominator. Thus, to divide 6/7 by 3, we may divide the numerator, giving 2/7, or we may multiply the denominator, which gives 6/21, which is equal to 2/7.

We will now multiply both terms of the fraction by the same number. Multiplying both terms of the fraction 2/3 by 3, we have 6/9. Here the denominator, expressing the number of parts into which the unit is divided, is three times as great as it was before; consequently each of the parts is only one third as great; but the numerator has also been multiplied by three, sot that three times as many parts are taken, and this makes the value of the fraction just equal to what it was before. So we may multiply by any number whatever, both terms of the fraction 2/3, and the value will still be the same as before; for example, 4/6, 6/9, 8/12, 10/15, 12/18, each of which is equal to 2/3. We may then at any time multiply both terms of a fraction by the same number, without altering the value of the fraction. By the same reasoning we may divide both terms of a fraction by the same number without altering its value. Taking the examples above, we may divide the terms of 4/6 by 2, and we obtain 2/3; dividing the terms of 6/9 by 3 gives us 2/3, and so the others; 2/3 is the same fraction as 4/6, 6/9, 8/12, etc., but it is expressed in lower terms, and therefore is more convenient. It is easier to write ½ than it is to write 16/32, though both have the same value.

To reduce a fraction to its lowest terms, we divide both the numerator and denominator by their greatest common divisor.

To find the greatest common divisor, separate each term into its prime factors, and erase those which are common to both. The remaining factors will express the value of the fraction in its lowest terms.

Treating the above fractions in this way they appear thus, , , , , , leaving in each case 2/3.

In how many ways can you obtain the answer to the following questions? 3/4x2? 5/6x3? 3/8x4? 3/14x3?

In how many ways can you obtain the answer to the following? 6/7÷3? 8/9÷4? 9/10÷3? 14/15÷2? 12/17÷4?

In how many ways can you obtain an answer to the following? 5/8÷2? 6/7÷4? 7/8÷3? 9/10÷2? 12/13÷5?

Reduce to their lowest terms each of the following fractions, 6/8, 14/21, 3/33, 20/32, 40/60, 15/50, 18/26, 24/27, 26/38, 6/16, 9/15, 15/33, 8/26, 30/46, 70/95.

TO FIND THE DIVISORS OF NUMBERS

Reduce the fraction to its lowest terms?

You will not see immediately that these two numbers have any common divisor. To assist you to reduce fractions of this kind, something will here be said about the way of finding the divisors of numbers. Let us first inquire what numbers can be divided by 2.

We have seen that all even numbers, and only those, can be divided by two.

What numbers can be divided by 4?

If you examine you will find that all even tense are divisible by 4, as 20, 40, 60, &c. If, therefore, the tens are even, and the units are divisible by 4, then the whole is divisible by 4. But the only unit numbers divisible by 4 are 4 and 8; therefore if the tens are even, and the unit number is 4 or 8, the whole is divisible by 4; as 84, 88, 124, 128, 148, 364, &c.

Again, as 10 when divided by 4 leaves a remainder of 2, any odd number of tens will do the same, as 30, 50, 70, 90; for every odd number of tens is an even number of tens +10. If, then, the number of tens is odd, the units must be two less than 4 or 8, in order to be divisible by 4. That is, if the tens are odd, and the units 2 or 6, the whole is divisible by 4; as 72, 96, 52, &c.

Are the following even numbers divisible by 4 or only by 2; and why? 126, 82, 94, 92, 138, 156, 346, 548, 76, 58, 392.

What numbers can be divided by 8?

As 1000 divided by 8 leaves a remainder of 4 (8x12=96,) it follows that 200 will be exactly divisible by 8, for the two remainders of 4 will make 8. If 200 is divisible by 8, it follows that all even hundreds are divisible by 8; as 400, 600, 1400, &c.

If, therefore, the hundreds are even and the tens and units are divisible by 8, the whole number will be divisible by 8; as 248, 672, 1456, &c.

Again, if the hundreds are odd and the tens and units are 4 less than some multiple of 8, for the odd hundred, divided by 8, leaves a remainder of 4; and this, added to the tens and units, will make an exact multiple of 8.

Are the following numbers divisible by 8; or by 4; and why: 444,944, 136, 1328, 712, 532, 816, 516, 384, 128, 1236.

What numbers are divisible by 5? All tens are divisible by 5; consequently if the unit figure is 5 or 0, the whole number is divisible by 5.

What numbers are divisible by 3? By examining the multiples of 3 we shall find this singular fact, that the sum of figures which express any multiple of 3 is itself a multiple of 3. Take the multiples of three from 12 to 24; 12, 15, 18, 21, 24; by adding the figures which express any one of these multiples we find that the sum is a multiple of 3. The figures of 12 added are 1+2=3, of 15 are 1+5=6, of 18 are 1+8=9, of 21 are 2+1=3, of 24 are 2+4=6. The same is true of all multiples of 3.

It will also be found that if you add the figures of any number and the sum is a multiple of three, the whole number is a multiple of three. To know, then, if a number is a multiple of 3, add together the figures that express the number, and if the sum is a multiple of 3, the whole number is a multiple of 3.

Are the following numbers divisible by 3? 471, 59, 115, 642, 624, 138, 234, 742, 894.

It follows from what has been said, that if any number is divisible by 3, and any other number expressed by the same figures differently arranged will also be divisible by 3; for the sum made by adding the figures will be the same in whatever order they are taken.

Thus, if 936 is divisible by 3; 369, 396, 963, 639, 693 are each divisible by 3.

We will next inquire what numbers are divisible by 6. As 6=2x3, any number that is divisible by 2 and by 3 is divisible by 6. You have learned what numbers are divisible by 3, and what by 2. If a number combines both these conditions, it is divisible by 6; that is, all numbers are divisible by 6, the sum of whose figures is a multiple of 3, and whose last figure is an even number.

What combinations of the figures 1, 2, 3, will give numbers divisible by 6; and what by 3 only.

Next let us inquire what numbers are divisible by 9.

If the figures which express any multiple of 9, as 18, 27, 36, 45, 54, be added together, the sum will be a multiple of 9.

Also, if the figures of any number be added together, and the sum is a multiple of 9, the whole number is divisible by 9.

Are the following numbers divisible by 9? and why? 936, 972, 396, 423, 387, 527, 411, 416, 315, 756.

Any number divisible by 9 and by 2 is divisible by 9x2, or 18; which of the above numbers are divisible by 18?

Any number divisible by 9 and by 4 is divisible by 9x4, or 36; which of the above numbers is divisible by 36?

Any number divisible by 9 and by 8 is divisible by 9x8, or 72; is either of the above numbers divisible by 72?

Any number divisible by 9 and by 5 is divisible by 9x5, or 45; which of the above numbers is divisible by 45?

What are divisors of 124? of 176? of 252? of 384? of 153? of 186? of 207? of 702? of 4041?

We will now return to the fraction that was first given. Reduce to its lowest terms.

Reduce to lowest terms, ; ; .

Reduce to lowest terms, ; ; .

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SECTION IX.

MULTIPLICATION OF FRACTIONS BY FRACTIONS.

We have seen how we may multiply or divide a fraction by a whole number. We will now inquire how we can multiply or divide one fraction by another. Let us multiply 3/7 by 2/5. First multiply 3/7 by 2, which gives 6/7 for the answer. But here we have multiplied by 2, instead of the real multiplier, 2/5. Now 2 is 5 times greater that 2/5; the product of 6/7 then is 5 times greater than it should be. It must therefore be divided by 5. We divide 6/7 by 5 by multiplying the denominator by 5, giving 6/35 for the answer.

DIVISION OF FRACTIONS BY FRACTIONS.

Let us now divide 4/5 by 3/7. First divide 4/5 by 3. This we do by multiplying the denominator by 3, giving the answer 4/15. Here, however, we have divided by 3, instead of the true divisor 3/7. We have used a divisor seven times too large. The quotient, therefore, will be seven times too small; 4/15 must therefore be multiplied by 7, making the answer 28/15. In the same way perform the following: . . . . .

The above analysis shows the grounds of the rules usually given in Arithmetics for the multiplication and division of fractions.

For Multiplication, multiply the numerators together for a new numerator, and the denominators for a new denominator.

For Division, invert the divisor and proceed as in multiplication.

Sometimes we wish to find the value of a compound fraction, as 2/3 of ¾; in such case we may understand the sing of multiplication, x, to stand in the place of the word of, and treat it as a case of multiplication. for in the above example it is plain that one third of ¾ is 3/12, and two thirds is twice as much, that is 6/12.

What is 2/3 of ¾ of 4/9? Multiplying as we have done above, we have for the answer 24/108. But this operation may be shortened. We see that 4 appears as a factor both in the numerator and the denominator. We may then cancel them both, which will have the same effect as dividing both terms of the answer by 4. Again, 3 appears in both the numerator and the denominator, for in the denominator it is a factor of 9. We may therefore cancel 3 in both terms.

The question will then appear thus, , substituting 3 in place of the 9. Multiplying together the terms that now remain, we have 2/9 for the answer. This is the same fraction as 24/108. If you separate the terms of 24/108 into their prime factors, and cancel what are common to both, the remaining factors will give the fraction 2/9.

Multiply the fractions , writing the terms that are composite in the form of their prime factors, and canceling factors that are common in both, it will stand , which gives 1/18.

Multiply . .

Multiply . . .

TO MULTIPLY OR DIVIDE WHOLE NUMBERS BY FRACTIONS.

The above examples will show how to multiply or divide a whole number by a fraction.

Multiply 7 by 4/5. Multiplying 7 by 4 gives 28, which is five times too great, because 4 is five times greater than 4/5. We must therefore divide the answer by 5, thus 28/5. As this is more than 1, we can reduce it to a whole number and a fraction. As 5/5 is equal to one, 25/5 will be equal to 5; 28/5 therefore is equal to 5 3/5.

In this way multiply 6 by 7/8. 9 by ¾. 8 by 5/7.

This operation is in fact the same as multiplying a fraction by a whole number, which has been treated of already.

Let us next divide 7 by ¾. Dividing 7 by 3 we have 7/3; here, however, we have divided by a number 4 times too great, for 3 is four times greater than ¾, If the divisor is 4 times too great, the quotient will be 4 times too small; 7/3, therefore, must be multiplied by 4, giving 28/3 for the answer.

Divide 8 by 5/7. . 11÷3/7. 10÷3/11.

To reduce an improper fraction, as 13/3, to a whole number and a proper fraction, we have only to consider how many whole ones the fraction is equal to, and how much remains. Thus 12/4 is equal to 3; 13/4, therefore, is equal to 3 ¼.

Reduce 8/3, 17/5, 12/7, 36/5, 42/8, 56/9, 31/14, 91/12, 87/5.

In like manner, if we have a whole number and a fraction, we may always reduce it to an improper fraction.

ADDITION AND SUBTRACTION OF FRACTIONS.

Suppose we wish to add together 3 ½ so that its value shall be expressed in a single expression; we must change 3 to halves, which will be 6/2; adding 12 to this we will have 7/2 for the answer.

In order to unite separate numbers into one expression, they must be of the same kind. We cannot unite 2 bushels and 3 pecks in one expression. It is still 2 bushels and 3 pecks, and we can make nothing else of it; but if we change the bushels to pecks, making 8 pecks, we can then add the 3 pecks, and bring it all into one expression, 11 pecks. So to unite 5 2/3 we must change the 5 to thirds, making 15/3, and add the 2/3, making 17/3. This is called reducing a mixed number to an improper fraction.

Reduce to an improper fraction 7 1/3, 8 ¼, 4 3/7, 5 ½, 6 ¼, 9 ½, 3 2/3, 5 3/7, 15 ½, 16 ¾, 13 2/3, 20 ¾, 21 ¹/₅.

Supposing we wish to add ½ to ¼, we must change the ½ to fourths, making 2/4; adding these, we have ¾ for the answer.

Add ½ to 1/12, ½=6/12, 6/12+1/12=7/12, ans.

Add 1/7 to 3/14, 1/3+4/15. 1/5+4/20. 4/7+3/28, 5/6+17/18.

Let us now add 2/3 and 4/5. This question you perceive has a difficulty which the former ones had not; for 2/3 is no number of fifths, and therefore we cannot bring the fraction into fifths by any multiplication. we want a number for the denominator which can be divided both by 3 and 5. Now if you examine, you will find no such number until you come to 15. This, is of course, divisible by 3 and by 4, for those are its factors. We will then take 15 for the denominator. This we call the common denominator. Taking now the fractions 2/3 and 4/5, and changing the denominator 3 to 15, we see that we have made it 5 times as large as it was before; that is, we have multiplied it by 5. We must therefore multiply the numerator by 5, to preserve the value of the fraction. The fraction 2/3 then becomes 10/15 without altering its value. Passing now to the second fraction, 5/5, we see that in changing the denominator to 15, we must have multiplied it by 3; we must therefore multiply its numerator by 3. This will make the fraction 12/15. The two fractions will stand, then 10/15+12/15, which when added together are 22/15=1 7/15.

TO FIND A COMMON DENOMINATOR.

We can always obtain a common denominator, by multiplying all the denominators together; then, for the numerators, consider, in the case of each fraction, what its denominator has been multiplied by, in order to change it to the common denominator, and multiply the numerator by the same number. Thus each fraction will have had its numerator and its denominator multiplied by the same number, and so its value will not be changed.

What is the value of ½+⁴/₇? Of ²/₃+³/₄? Of ⁶/₇+⁴/₅? Of ½+²/₃? Of ¹/₅+²/₉? Of ⁶/₇+²/₉? Of ⁸/₉+³/₄?

Supposing we wish to add the fractions ¾ and 5/6. We can proceed as above, and with the common denominator, 24, the fraction will be 18/24+20/24. But we need not employ so large a denominator as 24. We seek the smallest denominator that shall contain both 4 and 6 as a factor. If now we separate 4 and 6 into their prime factors, we shall find the factor 2 belonging both to 4 and to 6; thus, 2x2, 2x3. Now one of these may be canceled, and we shall stile have 2x2 for the number 4, and 2x3 for the number 6. Multiplying the factors which remain, 2x2x3, we have 12 for the smallest common denominator.

From this we see, that, when both the denominators contain the same factor, we may reject it from one of them, and multiply together the factors that remain.

Add 3/8 to 5/12. Here 2x2 is common to both denominators, rejecting it in one, and multiplying, we obtain 24 for the least common denominator.

Add 5/18 to 6/27. Here 3x3 is common to both denominators, rejecting it in one, and multiplying, we obtain 54 for the least common denominator.

Add 17/20 to 5/32. Add 3/8 to 1/36. Add 4/15 to 9/25.

When more fractions than two are to be added it is often most convenient to add two together first, and then add a third to the sum of these, and so on.

Add 2/3+1/6+3/4. First add 2/3 and 1/6, which equals 5/6. Next, 5/6+3/4; 5/6=10/12, and ¾=9/12; 10/12+9/12=19/12=1 7/12, Ans.

Add 1/5+2/3+3/10. First add 1/5 and 3/10; then to the sum of these add 2/3.

Add 2/9+1/12+5/6. Add 3/5+2/3+4/15. Add 1/7+3/4+3/8.

Add 3/13+7/9. Add 8/11+3/7. Add 7/12+5/24+1/10.

From 11/12 subtract 5/12. From 5/6 sub. 7/12. From 11/16 sub. 5/32.

From 18/19 sub ¾. From 20/23 sub. 7/10. From 31/35 sub. 9/12.

Miscellaneous Examples.

1. A man spends 1/7 of a dollar in a day; what part of a dollar will he spend in 5 days?

2. A man earns 8/9 of a dollar in a day; how much can he earn in half a day? How much in 1/4 of a day? How much in 1/8 of a day?

Here consider whether you can divide the numerator.

3. A man earns 7/8 of a dollar in a day; how much can he earn in half a day? How much in 1/4 of a day? How much in 1/8 of a day?

Consider whether you can divide the numerator; and if you cannot, what you must do.

4. A vessel filled with water leaks so that 3/8 of its contents will leak out in a week; at this rate, what part will leak out in a day?

What is 1/7 of 3/8?

5. If a team ploughs 5/7 of an acre in 6 hours, how much will it plough in one hour? How much in 3 hours?

What is 1/6 of 5/7? What is 1/2 of 5/7?

6. If a horse runs 1/3 of a mile in one minute, how far will he run in 3/4 of a minute?

How far will he run in 6/7 of a minute?

What is 3/4 of 1/3? What is 6/7 of 1/3?

7. A man has 7/8 of a dollar, which he wishes to distribute equally among several persons, giving 3/16 of a dollar to each; how many can receive this sum, and what will be the remainder?

How many times is 1/16 contained in 7? 3/16 in 7? 3/16 in 7/8?

How many times is 1/15 contained in 4? 2/15 in 4? 9/15 in 4/7?

How many times is 1/20 contained in 6? 7/20 in 6? 7/20 in 6/7?

8. A man gave 5/7 of a bushel of oats to some horses, giving to each 1/8 of a bushel; to how many did he give it? And what was the remainder?

How many times will 1/16 go in 5? In 5/7? How many times will 9/16 go in 5/7?

9. A man has 7/8 of a dollar; he gives 1/4 of a dollar to one person, and 9/5 of a dollar to a second, what part of a dollar has he left?

How many cents had he at first? How many cents did he give away? How many cents had he left?

10. If 13 pounds of figs cost 9/8 of a dollar, what is that a pound?

11. If 5 1/2 lbs. Of figs cost 5/12 of a dollar, what is that a pound? Find first what one half pound will cost.

12. If 6/7 of a cwt. Of iron cost 4 1/3 dollars, what will a hundred weight cost?

13. If 34 1/2 lbs. Of tea cost 11 3/8 dollars, what will 1 pound cost?

Here you find 69/2 pounds cost 91/8 of a dollar: therefore 69 pounds must cost 91/4 of a dollar.

14. If 2/3 of a barrel of flour cost 3 3/8 dollars, what is that a barrel?

15. If wood is 5 1/2 dollars a cord, what will 7/16 of a cord cost? What will 4 1/2 cords cost?

16. If 33 1/2 gals. Of molasses cost 11 3/8 dollars, what is that a gallon?

17. If 31 1/2 gals. Of vinegar cost 4 5/8 dollars, what is that a gallon?

18. If a bottle of wine containing 1 1/2 pints cost 4/8 of a dollar, what would a barrel of wine come to at that rate?

19. In a pile of wood there is a 13 1/2 cords; how many loads of 3/4 of a cord each are there in the pile?

20. How many times will 2 1/4 go in 7 1/2? In 9 1/4? In 11?

21. How many loaves, of 8 1/2 oz. Of flour each, can be made from 7 pounds of flour?

22. If a family consume 3/12 pounds of flour a day, how long will a barrel of flour, that is 196 pounds, last them?

23. If a barrel of flour last a family 40 days, how long will 14 pounds last them?

24. A garrison of 100 men is allowed 12 oz. of flour a day to each man; how long will 10 barrels last them?

25. Two men hire a horse, a week, for 5 dollars; one travels with him 30 miles, the other 45 miles; what ought each to pay?

26. Two men hire a pasture in common for $4.80; one pastures his horse in it 7 1/2 weeks; the other pastures his horse 9 weeks; what ought each to pay?

27. A boy bought 3 doz. of oranges for 37 1/2 cents, and sold them for 1 1/2 cents apiece; what did he gain?

28. A man bought 7 yds. of cloth for 16 dollars, and sold it for 3 dollars a yard; what did he gain on each yard?

29. a man worth 1690 dollars, left 2/5 of his property to his wife; how much did she receive? The remainder he divided equally among 3 sons; what did each one receive?

30. A man bequeathed his estate of 14,000 dollars, one third to his wife, and the remainder to be divided equally among four sons; what did the wife and what did each son receive.

31. In an orchard one third of the trees bear apples, tow fifths as many bear plums, and the rest bear cherries; what portion of the trees bear plums? What portion bear cherries/ The number of cherry trees is 40; what is the whole number of trees in the orchard?

32. What is 3/7 of 549? What is 8/9 of 374?

33. What is 1/3 of 175 1/2? What is 5/6 of 198?

34. What is 1/2 of 3/5 of 1640? What is 2/3 of 972?

35. If 2 barrels of flour cost 11 1/2 dollars, what will 17 barrels cost? What will 22 1/2 barrels cost?

36. If 2 1/2 cords of wood cost 15 dollars, what will 68 3/4 cords cost? What will 200 cords?

37. If a horse eat 2 1/4 tons of hay in 30 weeks, what part of a ton will he eat in 1 week?

38. What is the cost of 23 1/2 yds. of cloth at 7/8 of a dollar a yard?

39. What is the cost of 31 1/2 gallons of molasses at 5/16 of a dollar a gallon?

40. A grocer drew from a cask containing 31 1/2 gallons, 14 of its contents. Now how much did he draw out? How much remained?

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SECTION X.

THE LEAST COMMON MULTIPLE.

The method stated in the foregoing section for finding the smallest common denominator, serves to introduce a topic which requires some more extended and careful study.

It often becomes desirable to ascertain, with respect to several numbers, what number there is which contains them all in itself as factors. A number which contains another number as a factor of itself is a multiple of that number. Thus 6 is a multiple of 2, and also of 3. a number which contains several numbers as factors of itself is a common multiple of those numbers. Thus 12 is a common multiple of 2 and 3.

The smallest number which contains several numbers as factors of itself, is the least common multiple of those numbers. Thus, though 12 is a common multiple of 2 and 3, it is not he least common multiple; for 6 contains them both as its factors; 6 is therefore a smaller common multiple of 2 and 3 than 12 is; and as no number smaller than 6 does contain 2 and 3 as its factors, 6 is the smallest common multiple of 2 and 3.

Suppose now we wish to find the smallest common multiple of 3 and 4. The number, it is clear, must be a certain number of 3s, and also a certain number of 5s. Now by multiplying 3 and 5 together we evidently obtain such a number; for it will be 3 times 5, and it will be 5 times 3. Multiplying the two numbers together then, will always give their common multiple. The next question is, will this product of the two numbers be their least common multiple? This will depend on the character of the two numbers. If the numbers are prime to each other their product will be their least common multiple. For example, in the numbers 3 and 5, if we take nay number of 5s less than 3, as 2x5, the factor 3 has disappeared, and the number is no longer a multiple of 3. If we take any number of 3’s less than 5, as 4x3, the factor 4 has disappeared, and the number is no longer a multiple of 5. The product, therefore of numbers prime to each other, is their least common multiple. In the above example, the numbers of 3 and 5 were prime in themselves, and not merely prime to each other. To make the principle more clear, we will that two numbers that are not prime in themselves, but are only prime to each other.

What is the least common multiple of 8 and 9? Multiplying them together we have 72. 72 is, then, a common multiple of 8 and 9. The question is, is it their smallest common multiple? Writing the numbers with their factors they are 2x2x2 and 3x3. Now if we erase one of the 2’s we have no longer the factors of 8, and the product of the factors will not be divisible by 8. In the same way, if we erase one of the 3’s the product will not be divisible by 9.

If, then, the numbers are either prime, or prime to each other, the product is their least common multiple.

Next let us inquire, what is the least common multiple of 4 and 6? Their product is 24, but this is evidently not their least common multiple, for 12 contains both 4 and 6 as factors. To show why it is, that in this case, something less than the product of the numbers is their least common multiple, we will express each by its factors, thus 2x2, 2x3. Now it is clear that any number of times which you take 2x2 as a factor will be a multiple of 2x2. If then we throw-out the 2 in the 2x3, and multiply by the remaining 3, the product will be a multiple of 2x2, or 4. Looking now at the 2x3, or 6, it is evident that any number of times which you may take that as a factor will be a multiple of 2x3. But the 2 we may take form the 2x2, throwing away that in the 2x3; this leaves us to multiply the 2x3 by 2; as we before multiplied the 2x2 by 3, making 1 as the least common multiple. The rule, therefore is: Retain of each prime factor the highest power which appears in any of the given numbers; erase the rest, and multiply together what then remain.

Find the least common multiple of 3, 24 and 36. Expressed by the factors they are 2x2x2. 2x2x2x3. 2x2x3x3. Now 2x2x2 is common to 8 and 36; we throw this out of 36, leaving 3x3. Finally 3, we find, is common to 24 and 36; throwing this out of 24, we find the numbers appear as follow: 2x2x2. 2x2x2x3. 2x2x3x3. These multiplied together give the least common multiple, 72. This conforms to the rule; for 2x2x2 is the highest power of the factor 2, and 3x3 of the factor 3. What is the least common multiple of 24, 60 and 100? These factors are 2x2x2x3; 2x2x3x2x5; 2x2x2x5x5. We see that 2x2 is common to them all; expunge it in the second and third number. Next, 3 is common to the 1^st and 2d; expunge it in the 2d. Lastly, 5 is common to the 2d and 3d; expunge it in the 2d, and the numbers will stand, 2x2x2x3. 2x2x3x5. 2x2x5x5. These multiplied together, give 600.

To multiply these most easily, first take 2x2x5x5=100; then the remaining factors, 2x3, multiplied by 100, give 600.

What is the least common multiple of 24, 40 and 72?

What is the least common multiple of 18, 54, 81?

What is the least common multiple of 15, 4, 7? Of 15, 40, 27? Of 16, 14, 6? Of 60, 12, 18?

From the foregoing reasoning and examples you will perceive that the least common multiple of several numbers is the product of all their prime factors, each taken in the highest power in which it appears in any of the numbers.

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SECTION XI.

PRACTICAL QUESTIONS.

1. What part of a shilling is 1 penny? 2 pence? 3 pence? 4 pence? 5 pence? 6 pence? 7 pence?

2. What part of a penny are 2 farthings? 3 farthings? 4 farthings? 5 farthings? 6 farthings? 8 farthings?

3. What part of a shilling is 1 farthing? 2 farthings? 3 farthings?

What part of a shilling is 1 penny and 1 farthing? 1 penny, 2 farthings? 3d 3qrs.? 4d 2 qrs.? 6d 1 qr.? 9d 2 qrs.?

4. What Part of a pound is 1 shilling? 2s.? 3s.? 5s.? 1s. 1d.? 2s. 1d.? 4s. 3d.? 5s. 6d.? 7s. 9d.? 3s. 8d.?

5. What part of a pound is 1 farthing? 2 qrs.? 3qrs.? 2d 3qrs.? 5d. 2 qrs.? 1s. 1d. 1 qr.? 6s. 7d. 3qrs.?

6. What part of a pound avoirdupois is 2 oz.? 3 oz.? 4 oz.? 5 oz.? 6 oz.? 7 oz.? 8 oz? 9 oz.? 10 oz.?

7. What part of one ounce is one dram? What part of 1 pound is one dram? 2 drs.? 3 drs.? 1 oz. 1 dr.? 1 oz. 2 drs.? 2 oz. 4 drs.? 3 oz. 6 drs.? 8 oz. 3 drs.? 9 oz. 11 drs.?

8. What part of a pound is 1/16 of an oz.? 3/16 of an oz.?

What part of a pound is ½ an oz.? 2 ½ oz.? of 3 ½ oz.? 4 ½ oz.?

9. What part of a pound Troy is 1 dwt.? 5 dwt.? 6 dwt.? 9 dwt.? 11 dwt.? 10 dwt.? 12 dwt.? 16 dwt.?

10. What part of an ell English is 1 qr. of a yard? 2 qrs.? 3 qrs.? What part of a qr. is 1 nail? 3 nails?

11. What part of a yd. is 1 qr. 1 nail? 2 qrs. 3 n.? 3 qrs. 2 n.? What part of an ell English is 3 nails? 1 qr. 3 n.? 4 qrs. 1 n.?

12. What part of a yd. is 1 inch? 4 inches? 7 inches? 9 inches? What part of a yard is 1 qr. 2 in.? 2 qrs. 3 in.? 3 qrs. 1 in.?

13. From a vessel containing 3 gallons of wine, 3 gills leaked out; what part of a gallon leaked out? What part of a gallon remained?

14. From a barrel full of wine 7 quarts were drawn; how many quarts remained? What part of a barrel had been drawn out? What part of a barrel had remained?

15. If 2/3 of a barrel of beer be divided into 4 equal parts, what part of a barrel will each of the parts be? How many gallons will each part be?

16. If one quart be taken from a barrel full of beer, what part of a barrel will remain? If 3 pints be taken out, what part will remain? If 7 ½ gallons be taken out, what part of a barrel is taken out? What part of a barrel remains?

17. A man distributed 7 ½ gallons of milk, and distribute it to some poor persons, giving 2/5 of a gallon to each, how many persons will you give it to? How much will remain?

19. What part of 1 foot is 1 ½ inches? 2 ½ in.? 5 ½ in.? 6 ¼ in.? 8 ¾ in.? 9 ¼ in.? 10 2/3 in.? 11 ¼ in.?

20. What part of a yard is 2 inches? 3 ½ inches? 14 in.? 5 ¼ in.? 6 ¼ in.? 17 ½ in.? 24 ¼ in.?

21. What part of a rod is ½ a foot? 1 ½ feet? 2 ½ feet? 4 ft. 3 in.? 6 ft. 7 in.? 10 ft. 5 in.?

22. What part of 3 rods is ½ a foot? 1 foot? 3 ½ feet? What part of a furlong are 2 ½ rods? 5 ½ rods?

23. What fraction of a foot is 1/7 of a yard? 2/7 of a yd.? What fraction of a foot is 1/13 of a rod? 2/13 of a rod? 3/13 of a rod?

24. A man measured the length of his barn with a stick half a yard long, and found the barn 31 ½ times the length of his stick; how long was it?

25. A carpenter is cutting up a board 17 ½ feet in length, into pieces 2 ¼ feet long; how many pieces will there be, and how long will be the piece that remains?

26. A man measures a piece of fence with a pole 9 ½ feet long; the fence is 15 ½ times the length of the pole; how many rods is it in length?

27. What part of a peck is 1/50 of a bushel?

What part of a gallon are 4/17 of a peck? ¾ of a peck?

What part of a quart is 1/22 of a peck? 2/23 of a peck?

What part of a quart are 4/75 of a bushel? 2/29 of a bushel?

28. What part of a peck is 1/8 of a bush.? 2/8 of a bush.? 3/8 of a bush.? 4/8 of a bush? 5/8 of a bush.?

29. Two men bought a lot of standing wood in company, for 11 dollars; one cut off 2 cords, the other 1 cord; what ought each to pay?

30. Two boys bought the chestnuts on a tree for 50 cents; one had 11 quarts, the other 6 quarts and 1 pint; what ought each to pay?

31. Three men bought a piece of cloth for 24 dollars? The first took 2 ½ yds., the second the same quantity, and on measuring the remainder it was found to be 3 yards; what ought each to pay?

32. Two men hire a horse for a month for 12 dollars; one travels 200 miles with the horse, the other 150; how much should each pay?

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SECTION XII.

DECIMAL FRACTIONS.

[See Numeration, Part II.]

In the calculations in common fractions, a great inconvenience arises from their irregularity. There is no law regulating the magnitude of either of the terms. The denominator may be in any ration whatever to the numerator. From seeing one you can make no inference at all respecting the magnitude of the other. In calculations of addition, it is often more than half the work to bring the fractions into a common denominator.

Now it is evident that if factions could be written in the same manner as whole numbers, that is, increasing in a ten fold rate as you advance to the left, and decreasing in a ten fold rate as you advance to the right, an immense gain would be made in the convenience of calculating them. Operations in fractions would then be just as easy a operations in whole numbers. Now this advantage is gained in decimal fractions. they are brought under the same law as whole numbers. Let us observe the manner in which whole numbers are written. Take the number 222; the right hand figure signifies two units, the next two tens, the next tow hundreds; just as if it were written in this manner, 2x100+2x10+2; two multiplied by 199 plus two multiplied by 10, plus two; making two hundred and twenty-two. But this cumbersome method of writing is unnecessary, because the law of notation determines what number the figures in each place shall be multiplied by. It must not be forgotten that the figure 2 in the above example in no case signifies of itself more than two. It is the place it occupies that gives it the higher value of tens or hundreds.

Now it would evidently be a great convenience if we could reduce fractions to the same law, so that they would, like whole numbers, decrease in a decimal ration, in advancing from the left tot he right. To show this regularity to the eye, we will right he following numbers: two multiplied by 1000, two multiplied by 100, and two divided by 1000. Written in full they would stand thus: 2x1000+2x100+2x10+2+2/10+2/100+2/1000.

6*

But we have seen that we may write the whole numbers without the multipliers, thus, 2222, because we know from the place each figure occupies what its multiplier must be. Just so we can write fractions without the denominators, provided we know, from the place of the numerator, what the denominator must be. Thus the whole of the above series may be written as follows; 2222.222. A decimal, therefore is the numerator of a fraction, whose denominator is never written, but is always understood to be 1, with as many ciphers as there are places in the decimal.

You observe that, in writing the series given above, there is a period placed at the right hand of the whole numbers, separating the unit figure from that of tenths. The period must never be omitted when there are fractions, for it enables you to determine the value of each figure in the sum. Instead of reading .22 two tenths and 2 hundreds, we may call it 22 hundredths, which is more convenient and amounts to the same; for 2 tenths is equal to 20 hundredths; so .222 is two hundred and twenty-two thousandths. So, in all cases, read the decimal numbers as whole numbers, and for their denominator take 1 with as many ciphers as there are places in the written decimals.

In all your study of decimals, be careful not to confound the words which express fractions with the similar words which express whole numbers; as tenths with tens, hundredths with hundreds. The following questions will aid you in fixing this distinction clearly in mind.

1. How many tenths are equal to ten whole ones?

2. How many tenths are equal to two and a half whole ones?

3. How many hundredths are equal to three and a quarter whole ones?

4. How many hundredths are equal to one hundred whole ones?

5. How many thousands are equal to ten whole ones?

6. In fifteen whole ones how many tenths? How many hundredths?

7. In seventy-five hundredths how many tenths?

8. In three tenths how many hundredths?

9. In six tenths how many thousandths?

Thus, you observe, fractions have been brought under the same law that regulates the writing of whole numbers. They may now be added, subtracted, multiplied, and divided, like whole numbers. But in doing this it is important to determine the place of the period that separates the whole numbers from the fractional part of the sum. Where must the period be placed in the answer?

ADDITION AND SUBTRACTION OF DECIMALS.

Let us first observe how important it is that the rule in this case be entirely correct. If I have this number, 32.5 to write, and by any mistake I should write it 3.25, it would denote a quantity only one tenth as great as it should be; or, if I should write 325. it would denote a quantity ten times greater than it should be. Moving the period one place to the right, makes the number ten times as great as it was before, for tens become hundreds, and hundreds, thousands; and each figure ten times as great as before. So, by moving the period one place to the left, the number becomes just one tenth what it was before. Removing the period two places from its true place, makes the number 100 times larger or smaller that it should be, according as you remove it to the right or the left. Hence you may see that in order to multiply a number that has decimals, by 10, you have only to remove the period one place to the right; to multiply by 100, remove the period one place on. To divide by 10, remove the period one place to the left; to divide by 100, remove it two places, and so on. From accurate in fixing the place of the decimal in the answer to any question.

We will begin with addition. Add 4.46 to 3.21. Here you observe the two whole numbers make 7, and 46 hundredths added to 21 hundredths make 67 hundredths: the answer, then must be 7.67, having two decimal places. Add 6.8 to 5.23. The 3 hundredths must evidently stand alone, since there is nothing like it to add to it; 2 tenths added to 8 tenths make 10 tenths, or one whole; this we carry to the 5, which gives us for the answer, 12.03. This will serve to suggest the rule for placing the period in the answer to questions in addition. The number of decimal places in the answer must be as great as can be found in any one of the numbers to be added.

The same rule holds in subtraction. Take for illustration the numbers given in the second example of addition. From 6.8 subtract 5.23. Now as inn the minuend there are no hundredths, we must borrow 10 in this place, and we shall have a remainder of 7 hundredths; adding 1 tenth to the subtrahend, to compensate for the 10 hundredths added to the minuend, we have in the place of tenths a remainder of 5; finally, in the place of units we subtract 5 from 6: the answer is 1.57. In performing this operation, you may, if you please call the 8 tenths 80 hundredths; then 23 hundredths from 80 hundredths leaves 57 hundredths. By performing slowly and with care examples of your own selection, you will see the verification of the rule given above, both for addition and subtraction.

Add 2.4 to 3.8. Add .6 to 1.3. Add .4 to .3. Add .37 to .25. Add 3.7 to 2.4. Add 1.08 to .05.

From 4.6 subtract 2.4. From 7.1 subtract 6.4. From .18 subtract .13. From 4.5 subtract .6.

In these examples each step should be explained by the pupil as he performs it.

MULTIPLICATION OF DECIMALS.

The rule in multiplication we shall find to be different from the above.

1. First, we will multiply 2.4 by 3. If we regard the multiplicand as a whole number, the answer will be 72. But by regarding the multiplicand as a whole number, ---as 24 instead of 2 and 4 tenths, ---we regarded it ten times greater than it really is; the answer, therefore, is ten times too great. Instead of 72 it must be 7.2.

2. Multiply 6.2 by 3.4. By regarding both as whole numbers we obtain the answer 2108. Now in calling the multiplicand 62 instead of 6.2 we treated it as 10 times too great, even if the multiplier were a whole number. We must therefore divide it by 10 or write 210.8. But the multiplier also is 10 times too great; the answer must therefore be divided again by 10, in order to bring it right. Thus the answer will stand 21.08.

3. Again; multiply .62 by 3.4. Here we obtain the same figures as before 2108; but by treating the multiplicand as a whole number, we regarded it as 100 times too great; the answer therefore must be divided by 100, or written 21.08. But the multiplier, calling it a whole number, was taken 10 times greater than it is; the answer must be again divided by 10, and thus it will stand 2.108.

4. Once more: multiply .62 by .34. The figures of the answer are as before, 2108, but by regarding both the factors as whole numbers, we take each 100 times greater than it is; we must therefore divide by 100 to correct the error in the multiplier, and again by 100 to correct the error in the multiplicand. This will remove the point four places to the left, and the true answer will be .2108. By examining these examples you will see that the pointing in each case conforms to the following rule.

Point off as many figures for decimals in the answer as there are decimal places in both the factors taken together.

5. Multiply 2.7 by .3. 6. Multiply .6 by .7. 7. Multiply 6 by .7. 8. Multiply .02 by .3. 9. Multiply .02 by .03. 10. Multiply .01 by .01.

DIVISION OF DECIMALS.

1. Divide 48 by 12. Ans. 4.

2. Divide 4.8 by 12. The figure expressing the answer is 4, as in the first case; but, observe, the dividend is only one tenth as large as before; the quotient, therefore is only one tenth as large. Instead of 4, it is .4.

3. Divide .48 by 12. The figure of the quotient is still 4, but as the dividend is only one-hundredth part as large as in the first example, the quotient will be only one-hundredth part of 4, or 4 hundredths, written thus .04.

4. Again: divide 48 by 1.2. The quotient is still 4, but we must investigate the question to see where this 4 must stand. You observe that the divisor is now only one tenth of 12. Now if the divisor is only one tenth as great as is it was before, you must consider how that will affect the quotient. You will perceive on reflection that as you diminish the divisor you increase the quotient. If you make the divisor half as great the quotient will be twice as great, and so proportionally of other numbers. Now as, in this instance the divisor is one tenth as great as before, the quotient must be ten times greater. The figure 4, then, which is the quotient figure, instead of standing in the place of units, as before, must stand in the place of tens; that is, it must be 40, the cipher merely showing that the 4 stands in the place of tens.

5. Once more: divide 48 by .12. Here again you have 4 for the quotient figure, for you can have no other; but on comparing this example with the first, you perceive the divisor is only one hundredth part as great; the 1quotient must therefore be one hundred times greater, that is, it is 4000, the ciphers merely removing the 4 into the place of hundreds.

On examining these examples carefully, you will see that each answer is unquestionably correct. “But by what rule,” you ask, “are these examples wrought?” They are not wrought by rule, but by reasoning on the numbers themselves; and the more you habituate yourself to reason in arithmetic the less need you will have to depend on rules.

With this suggestion I will now state a rule, which you may at any time follow, when you have not the time to look into the reason of the operation.

There must be as many decimals in the quotient as the decimals in the dividend exceed those in the divisor: when there are fewer decimals in the dividend than there are in the divisor, ciphers must be added so as to make the number equal.

We will now review the foregoing examples, and observe their conformity with the above rule. Example 1 has no decimals in the divisor or the dividend, therefore none in the quotient. Ex. 2, the dividend has one decimal, the divisor none.; the quotient has therefore one. Ex. 3, the dividend has two decimals, the divisor none; the quotient has two. Ex. 4, the dividend has none, the divisor one; there must then be a cipher added to the dividend, and then the quotient will be in whole numbers. Ex. 5, the dividend has none, the divisor two; there must then be two ciphers added and then the quotient will be in whole numbers.

6. Divide 45 by 15. Divide 4.5 by 15. Divide .45 by 14 Divide 45 by 1.5. Divide 45 by .15.

7. Divide 66 by 11. 6.6 by 11. .66 by 11. 66 by 1.1 66 by .11/

In calculations of Federal money, cents and mills are regarded as decimally; the point therefore separating the whole numbers from the fractions must be placed between the dollars and the cents. Thus 24.00 is 24 dolls.; 2.40 is 2 dolls. 40 cents; 0.24 is 24 cents.

8. A man divided $24.00 among 3 men; how much did each receive?

9. A man divided $2.40 among 3 men; how much did each receive? Divide 2.4 by 3.

10. A man divided $0.24 among 3 men; how much did each receive? Divide 2.4 by 3.

11. A man divided 36 dollars among 4 persons; how much did each receive? Divide 36 by 4.

12. A man divided $3.60 among 4 persons; how much did each receive? What is one fourth of $3.60?

13. A man divided $0.36 among 4 men; how much did each receive? What is one fourth of .36?

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SECTION XIII.

REDUCTION OF VULGAR FRACTIONS TO DECIMALS.

We have now seen that Decimal Fractions have this great advantage over Vulgar Fractions, -- that they conform to the same law of notation as whole numbers, and may be added subtracted, multiplied and divided in the same manner, and with the same ease as whole numbers. It is desirable therefore, to introduce them in the in a great many cases instead of Vulgar Fractions. The next question that arrives therefore is can a Vulgar Fraction be changed to a decimal, having the same value; and how can it be done? Take the fraction ½; we wish to reduce it to tenths; or in other words to express it in tenths. Now we can change any number to tenths by multiplying it by 10. Thus 3 is 30 tenths, 4 is 40 tenths. We will now take ½ and change the numerator 1 to tenths, and it will stand .10: but the fraction was not one, but one half of one; 10 therefore is twice as great as it should be; we must divided it, therefore, by 2; that is, by the denominator, and it will be .5. To reduce a vulgar fraction, then to a decimal: add a cipher to the numerator, and divide by the denominator. If one cipher is not enough to render the division complete, add more.

Reduce to a decimal 1/5; change the numerator to tenths; it will be .10, but the quantity be reduced to tenths was not one, but one fifth of one; 10, therefore, is 5 times greater than it should be; dividing by 5, the answer is .2.

Reduce to a decimal the fraction 4/5.

Reduce to a decimal the fraction 3/5.

Reduce to a decimal the fraction ¼.

Reduce to a decimal the fraction ¾.

I will here direct your attention to a fact that is interesting to notice. If the denominator of the vulgar fraction is one of the factors of 10, that is, if it is either 2 or 5, the decimal figure will be as many times the other factor as there are units in the numerator of the vulgar fraction. This will appear self-evident when we express the numbers by their factors. Thus in obtaining the decimal for ½ we divided 10 by 2; but 10 is 2x5 , therefore in dividing by 2, 2 we simply expunge the factor we divide by, and leave the other: 2) 2x5. So in the fraction 1/5, we obtain the decimal by dividing 10 by 5, which expunges the factor 5, 5) 5x2; in reducing 2/5 we divide 2x10 by 5, thus: 5)2x2x5; leaving twice the factor 2; in 3/5, 5)3x2x5, leaving 3 times the factor for 2; in 4/5, 5)2x2x2x5, leaving 4 times the factor 2.

2. We will now take the fraction ¼; preceding as before we wish to divide 10 by 4, thus 2x2) 2x5; here we see the division cannot be complete, for the divisor contains the factor 2 twice, while the dividend has it only once. If, however, we had multiplied the original numerator 1 by 100, instead of 10, we should have had 10 twice as a factor in the dividend, and of course each factor of 10 twice; 100 is 10x10, and 10 is 2x5. It would have stood then thus, 2x2)2x5x2x5; factor 2 as many times as the divisor has it. Expunging these we have remaining the factor 5 taken twice, or .25.

This process you may observe conforms to the rule, to add as many ciphers as may be necessary to render the division complete.

3. Reduce the vulgar fraction ¾ to a decimal. 30 is composed of the prime factors 3x2x5; it contains 2 only once, and therefore it is not divisible by 2x2; 30 must therefore be multiplied by 10. This will introduce another 2, and it will stand thus, 2x2)3x2x5x2x5. By expunging the two 2’s and multiplying together the other factors, we have .75 for the answer.

4. Reduce the fraction 1/8 to a decimal. 10 expressed by its factors is 2x5, and 8 is 2x2x2. We must therefore multiply 2x5 by 10 till it shall contain the factor 2 as many times as 8 contains the same factor. That is, the numerator 1 must be multiplied by a thousand. It will then stand, 2x2x2) 2x5x2x5x2x5. Expunging the three twos there remains the answer .125.

By examining the above examples you may observe this fact, that if the denominator of the vulgar fraction contains one of the factors of 10, that is, 2 or 5, one or more times as a factor, the decimal will contain the other factor, just as many times. Thus, ½ =.5, ¼ or 1/(2x2)=.25, or .5x.5; 1/8 or 1/(2x2x2)=.125 or .5x.5x.5. In the same way 1/5=.2; 1/25 or 1/(5x5)=.4 or .2x.2; 1/125 or 1/(5x5x5)=.008, or .2x.2x.2. In this way you may determine that 1/16, when reduced to a decimal will contain 5 four times as a factor, because 16 contains 2 four times as a factor. So 1/32 will contain 5 five times as a factor.

This is conveniently expressed by saying, whatever power of one of the factors of 10 the denominator of the vulgar fraction contains, the same power of the other factor will appear in the decimal.

5. Reduce 1/3 to a decimal fraction. Preparing the numbers as before, it will stand 3)2x5. You observe that 3 is different from either of the factors of 10. Now as 120 has only the factors 2 and 5, it is not divisible by 3 without a remainder.

If you add to the numerator ever so many ciphers, you will only increase the number of times that 2 and 5 appear in it as its factors, and the number can never become divisible by 3 without a remainder. The answer becomes .333+ and this indefinitely, as far as you may please to carry on the operation. On the same principle we shall find that it is not possible to express accurately in decimals any vulgar fraction whose denominator contains as a factor anything different from the factors of 10; for this denominator becomes, in the reduction, a divisor of 10 or some power of 10, and if it has anything in it as a factor which is prime to the factors of 10, the complete division is impossible. Thus 1/6 cannot be exactly expressed in decimals; because, though one of its factors, 2, is a divisor of 10, the other, 3, is prime to 10. On this principle the following questions may be examined.

can 1/7 be accurately expressed in decimals? Why?

Can 1/8 be accurately expressed in decimals? Why?

Can 1/9 be accurately expressed in decimals? Why?

Can 1/12 be accurately expressed in decimals? Why?

Can 1/20 ? 1/15 ? 1/24 ? 1/25 ? 1/30 ? 1/50 ? 1/90 ? 1/92 ? 1/75 ? 1/13 ? 1/14 ?

1/16 ? 1/35 ? 1/17 ?

6. Name all the denominators, from 2 up to 20, of such fractions as can be accurately expressed in decimals?

from 20 to 40? From 40 to 60? from 60 to 80?

7. Name all denominators, from 2 to 20, of such fractions as cannot be expressed accurately in decimals? From 20 to 40? From 40 to 60? from 60 to 80?

8. What is the value of 4 shillings expressed in the decimal of a £? As 1 shilling is 1/20 of a £, 4 s. is 4/20. We can change 4 to tenths by adding a cipher; it will then be 40; 4, however was not the number we wished to reduce to tenths, but 4/20; the answer, 40, is therefore 20 times too great; dividing by 20 it stands .2.. 4 shillings, then is 2 tents of a £.

9. Now reverse the operation; what is the value in shillings of .2 of a £? Now shillings are twentieths; we can change any number to twentieths by multiplying it by 20, as 1 is to 20 twentieths, 2 is 40 twentieths, &c. Multiplying the .2 by 20 we have 40; but observe the two was not two wholes, but two tenths; the answer, 40 therefore is ten times too great; dividing by 10 the answer is 4 shillings.

10. Reduce to decimal of a £, 2 shillings. 5 shillings.

11. What is the value in shillings of .1 of a £? of .25 of a £?

12. Reduce to a decimal of a shilling, 3 pence. 3 pence are 3/12 of a shilling; reducing to hundredths to render the division complete, the ans. is .25.

13. What is the value in pence of .25 of a shilling?

14. Reduce 9 pence to the decimal of a shilling.

15. Reduce 1 peck to the decimal of a bushel.

16. Reduce 3 pecks to the decimal of a bushel.

17. Reduce .5 of a bushel to pecks. .76 of a bu. to pecks.

18. Reduce 15 minutes to the decimal of an hour.

19. Reduce 45 minutes to the decimal of an hour.

20. Reduce to minutes .5 of an hour. .25 of an hour. .75 of an hour.

21. Reduce 6 in. to the dec. of a foot. 9in. to dec. of a ft. 3 in. to the dec. of a ft.

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SECTION XIV.

INTEREST.

Interest is the sum paid by the borrower to the lender for the use of money. The rate of interest is established by law, and varies in different countries. In England it is 5 per cent., that is, 5 for the use of 100 for 1 year; in the New England States it is 6 per cent.; in New York it is 7 per cent. When no particular rate is mentioned in this book, 6 percent, will be understood.

If I borrow 100 dollars for 1 year, at the end of the year I owe the sum I borrowed, 100 dollars, and 6 dollars for the use of it, making 106 dollars. The sum borrowed is the principal; the sum paid for the use of it is the interest; the principal and interest added together make the amount.

1. What is the interest of 100 dolls. for 2 years? 8 years? 4 years? 6 years? 6 years? 7 years?

2. What is the interest of 200 dolls. for 2 years? for 4 years? 5 years? 6 years?

3. What is the interest of 300 dolls. for 2 years? for 4 years? of 400 dolls. for 3 years?

4. What is the interest of 50 dolls. for 1 year? for 3 years? of 25 dolls. for 1 year? 2 years?

5. What is the interest of 100 dolls. for 1 year? What is the interest of 100 cents for 1 year? What is the interest of 2 dolls. for 1 year? of 3 dolls.? of 4 dolls.? 5 dolls.? 6 dolls.? 7 dolls.? 8 dolls.? 9 dolls.?

6. What is the interest of 36 dolls. for 1 year? of 47 dolls.? of 57 dolls.? 34 dolls? of 62 dolls? of 89 dolls? of 125 dolls.? of 136 dolls.? of 20-7 dolls.? of 361 dolls.>

7. What is the interest of 50 cents for 1 year? of 25 cents? of 10 cents? of 30 cents? of 40 cents? of 50 cents? of 70 cents? of 80 cents? of 90 cents?

8. What is the interest of 50 doll. 60 cents for 1 year? of 84.30? of 96.40? of 112.25? of 230.75?

9. What is the interest of 100 dolls. for 6 months? for 3 months? for 2 months? for 1 month? for 4 months? for 5 months? for 7 months? for 8 months? for 9 months? for 10 months? for 11 months?

10. What is the interest of 10 dolls. for 6 mo.? 3 mo.? 2 mo.? 1 mo.? 4 mo.? 5 mo.? 7 mo.? 9 mo.? 10 mo.? 11 mo.?

11. What is the interest of 1 doll. for 6 mo.? 1 mo.? The interest of 1 dollar for 1 month is half a cent, and for any number of months, is half as many cents.

12. What is the interest of 1 dollar for 5 months? 7 mo.? 8 mo.? 9 mo.? 11 mo.? 12 mo.? 15 mo.? 16 mo.? 17 mo.? 18 mo.? The interest on any number of dollars for 1 month is half as many cents.

13. What is the interest of 12 dollars for 1 mo.? of 15 dolls.? of 25 dolls.? 34 dolls.? 42 dolls.? 67 dolls.? 93 dolls.? 104 dolls.?

14. What is the interest of 12 dolls. for 3 months? What is the interest of 25 dolls. for 6 months? In computing interest a month is reckoned 30 days.

As the interest on a dollar for 30 days is half a cent, that is 5 mills, the interest on a dollar for 1 fifth of 30 days will be 1 mill. One fifth of 30 is 6; the interest therefore on 1 dollar for 6 days is 1 mill, and the interest on any number of dollars for 6 days will be as many mills as there are dollars.

15. What is the interest of 16 dollars for 6 days? of 25 dolls.? of 40 dolls.? of 65 dolls.? of 75 dolls.? of 100 dolls.? of 500 dolls.? of 360 dolls.? of 840 dolls.? of 1000 dolls.?

As the interest of 1 doll. for 6 days is 1 mill, for 12 days will be 2 mills, for 18 days 3 mills, &c.

16. What is the interest of 1 doll. fro 24 days? of 2 dolls. for 6 days? of 2 dolls. for 12 days? of 2 dolls. for 18 days? of 5 dolls. for 6 days? for 12 days? for 24 days? of 36 dolls. for 18 days?

17. What is the interest of 125 dolls. for 1 year and 6 mo.?

18. What is the int. of 268 dolls. fro 3 years 4 mo.?

19. What is the int. of 45 dolls. for 4 years 7 mo.?

20. What is the int. of 60 dolls. for 1 year 3mo. 18 days?

21. What is the int. of 100 dolls. for 2 years 1 mo. 12 days?

22. What is the int. of 165 dolls. for 3 years 2 mo. 6 days?

23. What is the int. of 50.45 for 1 year 7 mo. 12 days?

24. What is the int. of 94 dolls. for 8 mo. 24 days?

25. What is the int. of 132.25 for 6 mo. 3 days?

26. What is the int. of 81.20 for 4 months 15 days?

27. What is the int. of 64.50 for 10 months 16 days?

28. What is the int. of 86 dolls. for 9 days?

29. What is the int. of 340 dolls. for 15 days?

30. What is the int. of 875 dolls. for 22 days?

When interest is more or less than 6 per cent., first find the interest at 6 per cent. and then make a proportional addition or subtraction for the required per cent. If it is 7 per cent. add one sixth; if 5 per cent. subtract one sixth.

31. What is the int. of 140 dolls. for 1 year, at 7 per cent.?

32. What is the int. of 200 dolls. for 1 year and 6 mo. at 5 per cent.?

33. What is the int. of 460 dolls. for 1 year at 4 1/2 per cent.

Remark. -- 4 1/2 is three fourths of 6.

34. What is the int. of 500 dolls. for 1 mo. at 9 per cent.?

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BANKING.

When money is obtained at a Bank, the note which is given for it promises to pay it at a certain time, as 60, 90, or 120 days. The interest on this note, instead of being paid at the end of the time, when the note is taken up, is paid before hand; that is, it is subtracted from the sum named in the note. so that, when you take up the note, you have only to pay the face of it, as the interest has been paid already.

If you give a note to a Bank for 100 dolls. to be paid in 90 days, they subtract from the sum named in the note the interest of the sum of 90 days, and three days besides, called days of grace; the balance is the sum you receive. The interest of 100 dolls. for 90 days is $1.50; for 3 days it is 5 cents; $1.55 subtracted from $100.0 leaves a balance of $98.45, which is the sum you will receive.

If the note is given for 60 days, the intest is cast for 63 days, and subtracted from the sum named.

The interest thus subtracted is called the bank discount; and the bank, when it lends money on such a note is said to discount the note.

35. What is the bank discount on a note of 100 dollars payable in 30 days? and how much will be received on such a note?

tHE INTEST ON 100 DOLLARS FOR 30 DAYS IS 50 CENTS; FOR 3 DAYS IT IS 5 CENTS; THE DISCOUNT, 55 CENTS, SUBTRACTED FROM 100 DOLLARS, LEAVES $99.45, the sum received.

36. What is the bank discount on a note for 200 dollars for 60 days? and what is the cash value of the note?

37. What is the bank discount, and what is the cash value of a note for 150 dollars payable in 30 days?

38. What is the bank discount, and what is the cash value of a note for 200 dollars payable in 90 days?

39. What is the bank discount, and what is the cash value of a note for 300 dollars payable in 90 days?

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DISCOUNT.

When money is paid by the debtor before it becomes due, an allowance is made, which is called discount. If I owe 100 dollars, to be paid in three months from this time, and I pay it now, I ought note to pay the full hundred dollars, for I am entitled to the use of the money three months longer. The sum which should be paid now, to cancel a debt due at some future time, is called the present worth of the debt.

To find the present worth of a debt due at some future time, first find the interest on the debt from the time of payment to the time when the debt is due; subtract this interest from the debt, and the remainder will be the present worth. Thus; if I pay a debt of 100 dollars three months before it is due, I subtract the interest of 100 dollars for three months, == $1.50, -- from 100 dollars, leaving $98.50 for the sum which I must pay.

This rule is not strictly equitable, because $98.50, with three months' interest added, will not amount to $100. The above method, therefore, givers the present worth a little too small; but it is the method uniformly adopted in business, and the error is on the right side, for it encourages the debtor to be prompt in his payments.

40. What is the present worth of 200 dollars payable in 1 year?

41. What is the present worth of 150 dollars payable in 2 years?

42. What is the present worth of 60 dollars payable in 6 months?

43. What is the present worth of 530 dollars payable in 1 year?

What is the present worth of 400 dollars payable in 1 year and six months?

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LOSS AND GAIN. -- PER CENTAGE.

44. A boy bought a pen knife for 25 cents, and sold it for 28 cents; how many cents did he gain over a quarter of a dollar?

45. Suppose he had bought 4 knives at the same price each, and sold them at the same profit, he would then have traded with a dollar; how much would he have gained on a dollar?

46. A boy bought a bushel of apples for 50 cents, and sold them for 59 cents; how much did he make per cent?

47. A bookseller bought a book for 75 cents, and sold it for 84 cents; how much did he gain per cent.?

As 75 is 3/4 of 100, what he gained on the book will be 3/4 of what he would gain on a hundred; or what he would gain per cent.

48. A boy bought some melons for 40 cents, and sold them for 60 cents; what did he make per cent.?

Ans. His gain was equal to half his outlay.

49. A grocer bought a lot of flower for 5 dollars a barrel; but finding it damaged, he sold it for 4 dollars a barrel; what did he lose per cent.?

50. A man bought a share in a bank for 80 dollars, and sold it for 82 dollars; what did he gain per cent.?

51. A man bought a lot of apples for $1.50 a barrel; what must he sell them for to gain 10 per cent.?