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