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Q. Now, if we take, and multiply both the 2 and 4 by 2, we ob ; what, then, is equal to ?

A., or.

Q. Now it is plain that the reverse of this must be true; for, if we divide both the 4 and 8 in by 2. we obtain, and, dividing the 2 and 4 in by 2, we have; what, then, may be inferred from these remarks respecting multiplying or dividing both the numerator and denominator of the same fraction?

A. That they may both be multiplied, or divided, by the same number, without altering the value of the fraction.

Q. What are the numerator and denominator of the same fraction ralled?

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A. The terms of the fraction.

Q. What is the process of changing into its equal called?
A. Reducing the fraction to its lowest terms

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of an hour, and minutes are 18; make, reduced to its lowest terms?

Q. How do you get the example?

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A. By dividing 15 and 60 each by 5.
Q. How do you get the ?

A. By dividing 3 and 12, each, by 3.

Q. How do you know that is reduced to its lowest terms? A. Because there is no number greater than 1 that will divide

both the terms of without a remainder

From these illustrations we derive the following

RULE.

113

Q. How do you proceed to reduce a fraction to its lowest terms? A. Divide both the terms of the fraction by any number that will divide them without a remainder, and the quotients again in the same manner.

Q. When is the fraction said to be reduced to its lowest terms?

A. When there is no number greater than 1 that will divide the terms without a remainder.

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1. If 1 apple cost of a cent, what will 2 apples cost? How much is 2 times } ?

2. If a horse eat of a bushel of oats in one day, how many bushels will he eat in 2 days? In 3 days? How much is two times ? 3 times ?

3. William has of a melon, and Thomas 2 times as much; what is Thomas's part? How much is 2 times ? 2 times? 2 times ? 3 times ? 6 times?

Q. From these examples, what effect does multiplying the numera tor by any number appear to have on the value of the fraction, if the denominator remain the same?

A. It multiplies the value by that number.

Q. 2 times is; but, if we divide the denominator 4 (in †) by 2, we obtain ; what effect, then, does dividing the denominator by any number have on the value of a fraction, if the numerator remain the same?

A. It multiplies the value by that number

Q. What is the reason of this?

A. Dividing the denominator makes the parts of a whole so many times larger; and, if as many are taken, as before, (which will be the case if the numerator remain the same,) the value of the fraction is evidently increased so many times.

Again, as the numerator shows how many parts of a whole are taken, multiplying the numerator by any number, if the denominator remain the same, increases the number of parts taken; consequently, it increases the value of the fraction.

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4. At of a dollar a yard, what will 4 yards of cloth cost? 4 times are of a dollar, Ans. But, by dividing the denominator of by 4, as above shown, we immediately have in its lowest terms.

From these illustrations we derive the following

RULE.

Q. How can you multiply a fraction by a whole number? A. Multiply the numerator by it without changing its denominator.

Q. How can you shorten this process?

A. Divide the denominator by the whole number, when it can be done without a remainder

Exercises for the Slate.

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8-6 bushels. what will 205

1. If a horse consume of a bushel of oats in one day, how many bushels will he consume in 30 days? A. 2. If 1 pound of butter cost of a dollar, pounds cost? A. 15—3015—30 dollars.

3. Bought 400 yards of calico, at § of a dollar a yard; what did it come to? A. 1200-$150.

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14. At 2 dollars a yard, what will 9 yards of cloth cost? 9 times 2 are 18, and 9 times are =1, which, added to 18, makes 19 dollars. A. This process is substantially the same as ¶ XXVII., by which the remaining examples in this rule may be performed.

15. Multiply 34 by 367. A. 11924.

16. Multiply 63 by 211.

A. 1450§.

17. Multiply 3 by 42. A. 129}}=1294.

¶ XXXIX. TO MULTIPLY A WHOLE NUMBER BY A FRACTION.

Q. When a number is added to itself several times, this repeated addition has been called multiplication; but the term has a more extensive application. It often happens that not a whole number only, but a certain portion of it, is to be repeated several times; as, for instance, If you pay 12 cents for a melon, what will of one cost? of 12 cents is 3 cents; and to get, it is plain that we must repeat the 3, 3 times, making 9 cents, the answer; when, then, a certain portion of the muluplicand is repeated several times, or as many times as the numerator shows, what is it called?

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A. Multiplying by a fraction.

Q. How much is of 12? of 12? of 20? of 20? of 8 ? of 8 of 40? of 40? of 40? of 40?

Q. We found in Multiplication, ¶ X., that when two numbers are to be multiplied together, either may be the multiplier; hence, to multiply a whole number by a fraction, is the same as a fraction by a whole number; consequently, the operations of both are the same as that described in ¶ XXVII.; what, then, is the rule for multiplying a whole number by a fraction? (For answer, see ¶ XXVII.)

Exercises for the Slate.

1. What will 600 bushels of oats cost, at of a dollar a bushel? A. $112.

2. What will 2700 yards of tape cost, at of a dollar a yard? 4. $337.

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1 XL. TO DIVIDE A FRACTION BY A WHOLE NUMBER.

1. If 3 apples cost of a cent, what will 1 apple cost? How much is 3?

2. If a horse eator of a bushel of meal in 2 days, how much will he eat in one day? How much is ÷ 27 3. A rich man divided § of a barrel of flour among 6 poor men; how much did each receive? How much is §6? 4. If 3 yards of calico cost of a dollar, how much is it a yard? How much is 5. If 3 yards of cloth cost a yard?

3?

of a dollar, how much is it

The foregoing examples have been performed by simply dividing their numerators, and retaining the same denominator, for the following reason, that the numerator tells how many parts any thing is divided into; as, arc 4 parts, and, to divide 4 parts by 2, we have only to say, 2 in 4, 2 times, as in whole numbers. But it will often happen, that the numerator cannot be exactly divided by the whole number, as in the following examples.

6. William divided of an orange among his 2 little brothers; what was each brother's part?

We have seen, (¶ XXXVII.) that the value of the fraction is not altered by multiplying both of its terms by the same number; hence X2=8. Now, & are 6 parts, and William can give 3 parts to each of his two brothers; for 2 in 6, 3 times. A. of an orange apiece.

Q. In this last example, if (in ) we multiply the denɔminator, 4, by 2, (the whole number,) we have, the same result as before; why is this?

A. Multiplying the denominator makes the parts so many times smaller; and, if the numerator remain the same, no more are taken than before; consequently, the value is lessened so many times.

From these illustrations we derive the following

RULE.

Q. When the numerator can be divided by the whole number without a remainder, how do you proceed?

A. Divide the numerator by the whole number, writing the denominator under the quotient.

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