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1. A man bought 30 cows at 25 dollars each; he then exchanged the cows for horses at 50 dollars each; how many horses did he get? Ans. 15 horses.

2. A farmer sold 1500 bushels of wheat at 125 cents per bushel, and received in return barley at 75 cents per bushel. How many bushels of barley did he get?

Ans. 2500 bu. barley.

3. How many acres of land, at 25 dollars per acre, can be obtained for 5 houses and lots at 750 dollars each? Ans. 150 acres.

4. How many yards of flannel, three-quarters of a yard wide, will line a coat made of 3 yards of cloth six-quarters wide ? Ans. 6 yards.

5. Sold 320 acres of land at 60 dollars per acre, and invested the proceeds in other land at 40 dollars per acre; how many acres did I get? Ans. 480 acres.

6. Exchanged 432 pieces of cloth at 18 dollars each, for linen at 6 dollars a piece. How many pieces of linen did I get? Ans. 1296 pieces.

7. Sold a farm of 477 acres at 48 dollars per acre, and invested the returns of sale in another farm at $36 per acre. How many acres did I buy? Ans. 636 acres.

and bought

each.

How many Ans. 2745.

8. Sold 15 horses at 732 dollars each, sheep for the proceeds at 4 dollars sheep did I buy?

9. Multiply the following numbers: 12, 15, 27, 28, 32, and divide the product by 2, 3, 4, 5, 6, 7, 8, and 9.

Ans. 12.

FRACTIONS.

DEF. 1.-If any thing or any number is divided into equal parts, one or more of these parts is a Fraction of the whole; and all the parts constitute the whole.

2. If any thing or any number is divided into two equal parts, each part is called One-half of the thing or number, and is written; if each half is divided into two equal parts, the whole number of parts will be four, and the one-half has made two of those parts; hence ; thus, 3; if each half were divided into. three equal parts, the whole would contain 6 of those parts, and f, or

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REM.-The upper term of the fraction is called the numerator, and the lower term the denominator.

3. If any thing or any number is divided into three equal parts, one of the parts is called One-third, and is written; and two of the parts are called two-thirds (); if each of these parts is divided into two equal parts, the whole will be divided into six equal parts, and 1 = 1; } = 4; that is, if the numerator and denominator of a fraction are both multiplied or divided by the same number, the value of the fraction is not changed.

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REM. Each fraction that has had both terms multiplied, may be restored to the original by division.

4. If 20 is divided into two equal parts, each part is

10, and 10 is one-half of 20; 10 x 2 = 20; and it may be expressed 18; and if numerator and denominator is divided by 10, thus, 1818=1, and 20 = 10; 0 = 10, and 101; 640 8, and 10 = 80.

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COROLLARIES.

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1. A fraction is exemplified by division; the dividend is the numerator, the divisor is the denominator, and the quotient is the fraction thus formed.

2. The denominator indicates the number of parts into which a number is divided, and the numerator indicates the number divided.

3. As only like numbers can be directly added or subtracted, so only fractions of the same denominator can be directly added or subtracted.

4. As multiplying or dividing both terms of a fraction by the same number does not change its value, consequently fractions having different denominators are readily reduced to a common denominator.

Thus, and are readily reduced to sixths. As 6 is the least common multiple of 2 and 3, it is also the least common denominator of the fractions and ; ×, and ; then + = ; and

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DEFINITIONS.

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1. When a unit is divided into equal parts, any number of those parts less than the whole expressed fractionally, is called a Proper Fraction; as, §, 1, 1, 1o, etc., in each of which the numerator is less than the denomi,

nator, and the fraction less than one; the result is the same in division where the dividend is less than the divisor; as,, 14, etc.; but when the divisor is less than the dividend, as, 4, 50, 5, etc., it is then called an Improper Fraction, and the fraction is greater than one.

2. When the division of an improper fraction is performed, and the divisor is not exactly contained in the dividend, the quotient becomes a mixed number; thus, 4 = 11; 50 = 8%, and 15 104; that is, partly integral 그후 and partly fractional.

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EXEMPLIFICATION.

In the above 54; 5 is the dividend, 4 the divisor, and the quotient, and the product of the divisor and quotient must be equal to the dividend; thus, 4 × = 5, the dividend. And 6 x 50 50, and 7×15 = 75. COR.-A Fraction may be regarded as an unexecuted

division.

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REM.--Fractions are similar to concrete numbers, and the operations performed on them are also similar. The numerator of a fraction of any denominator may be multiplied or divided by any abstract number, and the product or quotient will have the same denominator as the original fraction.

× 3 = 18, and 6 dollars × 3 = 18 dollars, and & ÷ 3 = , and 6 dollars 3 2 dollars.

A fraction may be multiplied by itself, whereas a concrete number cannot be; but a concrete number may be divided by a like number, giving an abstract quotient; so, also, the numerators of fractions having a common denominator may be divided, giving abstract numbers for the quotient.

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ADDITION AND SUBTRACTION OF FRACTIONS.

In the addition and subtraction of abstract numbers, as we can only unite like orders; that is, units and units, tens and tens, hundreds and hundreds, etc.; and as onehalf and one-half make two halves, one-fourth and twofourths make three-fourths, and , 1-1 = 4

-}=}, etc.; so fractions must have a common denominator in order to be united by addition and subtraction.

In order to reduce fractions to a common denominator, find the least common multiple of the denominators, which is a common denominator, and then multiply each numerator by the quotient arising from this common denominator divided by its own denominator; then each numerator and denominator will be multiplied by the same number.

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REM. The denominator indicates the number of parts into which a number has been divided; the numerator is the number itself.

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