T 51.

FRACTIONS.

111

T 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.

36

6

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, 1 barrels. multiplied by 40? 120 X Ans. 233.

men?

9 men?

2. What is the product of

how much?

40

by 18. Ly 21.

36.

by 48.

by 60.

by

Note. When the multiplier is a composite number, the pupil will rooolloat, (¶ 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, 12 = 12, and 13 × 5 = 62=51z, Ans.

4. Multiply 52 by 7.

An. 401.

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 151, 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 918 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?
in 3 days, supposing he travel 12 hours each day?

Ans. to the last, 772 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 dollars, that is, of 36 dollars, implies that 36 be first divided into 4 equal parts, and then that 1 of these parts ne 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 object in multiplying by a fraction, whatever may be the multiplicand, 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 27, the same as 364 × 3 = 27. Hence, To multiply by a fraction, whether be a whole number or a fraction,—

30 X B

RULE.

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, (T 50,) and the result will be the same.

EXAMPLES FOR PRACTICE.

2. What will 40 bushels of corn come to at of a dollar per bushel? 40 X = how much?

3. What will 24 yards of cloth cost at of a dollar per yard? 24 X = how much?

4. How much is of 90?

of 369?

5. Multiply 45 by 7. Multiply 20 by

To multiply one fraction by another.

53. 1. A man, owning of a ticket, sold share; what part of the whole ticket did he sell? how much?

Fo of 45?

of his

of is

We have just seen, (T 52,) that, to multiply by a fraction, is to divide the multiplicand by the denominator, and to multiply the quotient by the numerator. divided by 3, the denominator of the multiplying fraction, (49,) is, which, multiplied by 2, the numerator, (51,) is, Ans. The process, if carefully considered, will be found to consist in multiplying together the two numerators for a new numerator, and the two denominators for a new denominator.

EXAMPLES FOR PRACTICE.

2. A man, having of a dollar, gave of it for a dinner; what did the dinner cost him?

Ans. dollar. 3. Multiply by . Multiply by . Product, 4. How much is of of of ? 晷

Note. Fractions like the above, connected by the word of, are sometimes called compound fractions. The word of implies their continual multiplication into each other.

Ans. 188=20.

When there are several fractions to be multiplied continually together, as the several numerators are factors of the new numerator, and the several denominators are factors of the new denominator, the operation may be shortened by dropping those factors which are the same in both terms, on the principle explained in T 46. Thus, in the last example, t, ,,, we find a 4 and a 3 both among the numerators and among the denominators; therefore we drop them, multiplying together only the remaining numerators, 2 x 714, for a new numerator, and the remaining denominators, 5 X 8 = 40, for a new denominator, making to, Ans. as before.

K*

=

5. of off of of off of how much? Ans. fo. 6. What is the continual product of 7, 2, 4 of 3 and 34? Note. The integer 7 may be reduced to the form of an improper fraction by writing a unit under it for a denominator, thus, 7. Ans. 211.

7. At of a dollar a yard, what will of a yard of cloth cost?

8. At 63 dollars per barrel for flour, what will of a barrel cost?

61; then 5×7=1= $2121, Ans. Ans. $61. 10. At $2 per yard, what cost 6ğ yards? Ans. $ 1429. 11. What is the continued product of 3, 4, § 1 of of ?

9. At & of a dollar per yard, what cost 7 yards?

of 2, 24, and Ans. 1.

54. The RULE for the multiplication of fractions may now be presented at one view :

I. To multiply a fraction by a whole number, or a whole number by a fraction,-Divide the denominator by the whole number, when it can be done without a remainder; otherwise, multiply the numerator by it, and under the product write the denominator, which may then be reduced to a whole or mixed number.

II. To multiply a mixed number by a whole number,—Multiply the fraction and integers, separately, and add their products together.

III. To multiply one fraction by another, Multiply together the numerators for a new numerator, and the denominators for a new denominator.

Note. If either or both are mixed numbers, they may first be reduced to improper fractions.

EXAMPLES FOR PRACTICE.

1. At $ per yard, what cost 4 yards of cloth ? 6 yds.?

yds.?

8 yds.?

20 yds.? Ans. to the last, $15.

how much will it take to keep 3 horses the same time?

7 horses?

13 horses?

Ans. to last, 37

tons. 4. What will 8 barrels of cider come to, at $3 per barrel?

5. At $142 per cwt., what will be the 6. A owned & of a ticket; B owned ticket was so lucky as to draw a prize of each one's share of the money ?

7. Multiply of 3 by 2 of 4.

8. Multiply 7

by 21.

cost of 147 cwt.? of the same; the $1000; what was

Product, Product, 15. Product, 24. Product, 1. Product, 3.

of 8, of 7, § of 9, Product, 20. Product, 5555558.

To divide a whole number by a fraction.

55. We have already shown ( 49) how to divide a fraction by a whole number; we now proceed to show how to divide a whole number by a fraction.

1. A man divided $9 among some poor people, giving them of a dollar each; how many were the persons who received the money? 9÷2 how many?

1 dollar is 4, and 9 dollars is 9 times as many, that is, 36; then is contained in 36 as many times as 3 is contained in 36. Ans. 12 persons.

That is,-Multiply the dividend by the denominator of the dividing fraction, (thereby reducing the dividend to parts of the same magnitude as the divisor,) and divide the product by the numerator.

2. How many times is contained in 8?

8÷용

= how

many?

OPERATION.

8 Dividend.

5 Denominator.

Numerator, 3)40

Quotient, 134 times, the Answer.

To multiply by a fraction, we have seen, (¶ 52,) implies two operations a division and a multiplication; so, also, to divide by a fraction implies two operations-a multiplication and a division.