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5. If 2 bushels of apples cost of a dollar, what is that per bushel?
1 bushel is the half of 2 bushels; the half of is f
Ans. dollar. 6. If 3 horses consume 1 of a ton of hay in a month, what will 1 horse consume in the same time?
are 12 parts; if 3 horses consume 12 such parts in a month, as many times as 3 are contained in 12, so many parts 1 horse will consume. Ans. of a ton. 7. If of a barrel of flour be divided equally among 5 families, how much will each family receive?
is 25 parts; 5 into 25 goes 5 times. Ans. of a barrel. The process in the foregoing examples is evidently dividing a fraction by a whole number; and consists, as may be seen, in dividing the numerator, (when it can be done without a remainder,) and under the quotient writing the denominator. But it not unfrequently happens, that the numerator will not contain the whole number without a remainder.
8. A man divided of a dollar equally among 2 persons; what part of a dollar did he give to each?
of a dollar divided into 2 equal parts will be 4ths.
Ans. He gave of a dollar to each. 9. A mother divided a pie among 4 children; what part of the pie did she give to each? 4 how much? 10. A boy divided of an orange equally among 3 of his companions; what was each one's share? ÷ 3 = how much?
11. A man divided of an apple equally between 2 children; what part did he give to each? divided by 2 = what part of a whole one?
is 3 parts: if each of these parts be divided into 2 equal parts, they will make 6 parts. He may now give 3 parts to one, and 3 to the other: but 4ths divided into 2 equal parts, become 8ths. The parts are now twice so many, but they are only half so large; consequently, is only half so much as 2. Ans. of an apple.
In these last examples, the fraction has been divided by multiplying the denominator, without charging the numerator. The reason is obvious; for, by multiplying the denominator by any number, the parts are made so many times smaller, since it will take so many more of them to make a whole
one; and if no more of these smaller parts be taken than were before taken of the larger, that is, if the numerator be not changed, the value of the fraction is evidently made so many times less.
T 49. Hence, we have Two ways to divide a fraction by a whole number :
I. Divide the numerator by the whole number, (if it will contain it without a remainder,) and under the quotient write the denominator.-Otherwise,
II. Multiply the denominator by the whole number, and over the product write the numerator.
EXAMPLES FOR PRACTICE.
1. If 7 pounds of coffee cost of a dollar, what is that per pound? ÷ 7 = how much? Ans. of a dollar. 2. If of an acre produce 24 bushels, what part of an acre will produce 1 bushel? ÷ 24 how much? 3. If 12 skeins of silk cost of a dollar, what is that a
skein? 10 ÷ 12:
Note. When the divisor is a composite number, the intelligent pupil will perceive, that he can first divide by one component part, and the quotient thence arising by the other; thus he may frequently shorten the operation. In the last example, 168 × 2, and § ÷ 8 , and ÷2 Ans. T
5. Divide by 12. Divide by 21. Divide
6. If 6 bushels of wheat cost $47, what is it per
§ by 24.
Note. The mixed number may evidently be reduced to an improper fraction, and divided as before.
of a dollar, expressing the fraction in its
lowest terms. (T-46.)
7. Divide $44 by 9,
8. Divide 129 by 5.
9. Divide 142 by 8.
10. Divide 1841 by 7.
Quot. of a dollar.
Note. When the mixed number is large, it will be most convenient, first, to divide the whole number, and then reduce the remainder to an improper fraction; and, after dividing, annex the quotient of the fraction to the quotient of
the whole number; thus, in the last example, dividing 1841 by 7, as in whole numbers, we obtain 26 integers, with 24 remainder, which, divided by 7, gives, and 26+ =26, Ans.
11. Divide 27861 by 6.
12. How many times is 24 contained in 76461?
13. How many times is 3 contained in 462}?
To multiply a fraction by a whole number.
¶ 50. 1. If 1 yard of cloth cost of a dollar, what will 2 yards cust? X 2 = how much?
2. If a cow consume of a bushel of meal in 1 day, how much will she consume in 3 days? × 3 = how much 3. A boy bought 5 cakes, at 4 of a dollar each; what did he give for the whole? X5 = how much? 4. How much is 2 times ? times?
§ by 2.
3 times ?
5. Multiply by 3. 6. If a man spend & of a dollar per day, how much will he spend in 7 days?
is 3 parts. If he spend 3 such parts in 1 day, he will evidently spend 7 times 3, that is, 2 in 7 days. Hence, we perceive, a fraction is multiplied by multiplying the numerator, without changing the denominator.
But it has been made evident, (T 49,) that multiplying the denominator produces the same effect on the value of the frac tion, as dividing the numerator: hence, also, dividing the denominator will produce the same effect on the value of the fraction, as multiplying the numerator. In all cases, therefore, where one of the terms of the fraction is to be multiplied, the same result will be effected by dividing the other; and where one term is to be divided, the same result may be effected by multiplying the other.
This principle, borne distinctly in mind, will frequently enable the pupil to shorten the operations of fractions. Thus, in the following example:
At of a dollar for 1 pound of sugar, what will 11 pounds cost?
Multiplying the numerator by 11, we obtain for the product §¿=¿ of a dollar for the answer.
¶ 51. But, by applying the above principle, and dividing the denominator, instead of multiplying the numerator, we at once come to the answer,, in its lowest terms. Hence, there are Two ways to multiply a fraction by a whole number :—
I. Divide the denominator by the whole number, (when it can be done without a remainder,) and over the quotient write the numerator.-Otherwise,
II. Multiply the numerator by the whole number, and under the product write the denominator. If then it be an improper fraction, it may be reduced to a whole or mixed number.
EXAMPLES FOR PRACTICE.
1. If 1 man consume of a barrel of flour in a month, how much will 18 men consume in the same time? Ans. to the last, 14 barrels. multiplied by 40? 720 X Ans. 234.
2. What is the product of
Note. When the multiplier is a composite number, the pupil will recollect, (T 11,) that he may first multiply by one component part, and that product by the other. Thus, in the last example, the multiplier 60 is equal to 12 X 5; therefore, TX 12, and 3×5==5+, Ans. Am 401.
4. Multiply 54 by 7.
Note. It is evident, that the mixed number may be reduced to an improper fraction, and multiplied, as in the preceding examples; but the operation will usually be shorter, to multiply the fraction and whole number separately, and add the results together. Thus, in the last example, 7 times 5 are 35; and 7 times are 41=51, which, added to 35, make 401, Ans.
Or, we may multiply the fraction first, and, writing down the fraction, reserve the integers, to be carried to the product of the whole number.
5. What will 948 tons of hay come to at $17 per ton? Ans. $164. 6. If a man travel 2 miles in 1 hour, how far will he
in 8 hours?
in 12 hours?
travel in 5 hours?
Ans. to the last, 773 miles.
Note. The fraction is here reduced to its lowest terms; the same will be done in all following examples.
To multiply a whole number by a fraction.
¶ 52. 1. If 36 dollars be paid for a piece of cloth, what costs of it? 36 X how much?
of the quantity will cost of the price; a time 36 dol lars, that is, of 36 dollars, implies that 36 be first divided into 4 equal parts, and then that 1 of these parts be taken 3 times; 4 into 36 goes 9 times, and 3 times 9 is 27.
Ans. 27 dollars. From the above example, it plainly appears, that the ob ject in multiplying by a fraction, whatever may be the multipli cand, is, to take out of the multiplicand a part, denoted by the multiplying fraction; and that this operation is composed of two others, namely, a division by the denominator of the multiplying fraction, and a multiplication of the quotient by the numerator. It is matter of indifference, as it respects the result, which of these operations precedes the other, for 36 × 3 ÷ 4 = 27, the same as 36 ÷ 4 × 3 = 27.
Hence, To multiply by a fraction, whether the multiplicand be a whole number or a fraction,
Divide the multiplicand by the denominator of the multiplying fraction, and multiply the quotient by the numerator; or, (which will often be found more convenient in practice,) first multiply by the numerator, and divide the product by the denominator.
Multiplication, therefore, when applied to fractions, does not always imply augmentation or increase, as in whole numbers; for, when the multiplier is less than unity, it will always require the product to be less than the multiplicand, to which it would be only equal if the multiplier were 1.
We have seen, (T 10,) that, when two numbers are multiplied together, either of them may be made the multiplier, without affecting the result. In the last example, therefore, instead of multiplying 16 by, we may multiply by 16, (50,) and the result will be the same.