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

15. Multiply 34 by 367. A. 11923.

16. Multiply 6 by 211. A. 1450§.

17. Multiply 3 by 42. A. 12988-1294.

¶ 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 multiplicand is repeated several times, or as many times as the numerator shows, what is it called?

A. Multiplying by a fraction.

of 20 of 20 of 8? of 40?

Q. How much is of 12? of 12? of 8 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? A. $337

¶ 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 2?

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 ÷3?

5. If 3 yards of cloth cost of a dollar, how much is it a yard?

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, are 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 &×2=g. 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 denominator, 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.

Q. When the numerator cannot be thus divided, how do you proceed?

A. Multiply the denominator by the whole number, writing the result under the numerator.

Exercises for the Slate.

16

1. If 8 yards of tape cost of a dollar, how much is it a yard? How much is 6÷8.

Note. When a mixed number occurs, reduce it to an improper fraction, then divide as before.

7. Divide $6 among five men. A. 63=4÷5=37=lgo.

I XLI. TO MULTIPLY ONE FRACTION BY ANOTHER.

1. A man, owning § of a packet, sells part of the whole packet did he sell? 3 x 5 = 15

4 X 8 = 32

Ans. The reason of this

from the following illustration :

of his part; what How much is of ?

operation will appear

Once is, and of is evidently divided by 4, which is done (¶ XL.) by multiplying the denominator, 8, by the 4, making 32; that is, of =2.

Again, if of

much, that is, 2.

be, then of § will be 5 times as

Again, if of be, then will be 3 times, Ans., as before.

The above process, by close inspection, will be found to

consist in multiplying together the two numerators for a new numerator, and the two denominators for a new denominator.

Should a whole number occur in any example, it may be reduced to an improper fraction, by placing the figure 1 under it; thus 7 becomes ; for, since the value of a fractior ̧¶ XXXIV.) is the numerator divided by the denominator, the value of is 7; for, 1 in 7, 7 times.

From these illustrations we derive the following

RULE.

Q. How do you proceed to multiply one fraction by another? A. Multiply the numerators together for a new numerator; and the denominators together for a new denominator.

Note. If the fraction be a mixed number, reduce it to an improper fraction; then proceed as before.

Q. What are such fractions as these sometimes called?

A. Compound Fractions.

Q. What does the word of denote ?

A. Their continual multiplication into each other.

Exercises for the Slate.

1. A man, having of a factory, sold

of his part; what part of the whole did he sell? How much is of? × 3 180=25, Ans.

2. At of a dollar a yard, what will of a yard of cloth cost? How much is of §? A. 76.

Note. If the denominator of any fraction be equal to the numerator of any other fraction, they may both be dropped on the principle explained in ¶ XXXVII.; thus of of may be shortened, by dropping the numerator 3, and denominator 3; the remaining terms, being multiplied together, will produce the fraction required in lower terms; thus, of ÷ of § = 2 of &=10=12, Ans.

The answers to the following examples express the fraction in its

lowest terms.

1 XLII. To find the LEAST COMMON MULTIPLE OF TWO OR MORE NUMBERS.

Q. 12 is a number produced by multiplying 2 (a factor) by some other factor; thus, 2×6= 12; what, then, may the 12 be called? A. The multiple of 2.

Q. 12 is also produced by multiplying not only 2, but 3 and 6, likewise, each by some other number; thus, 2X612; 3×4=12; 6X2=12; when, then, a number is a multiple of several factors or numbers, what is it called?

A. The common multiple of these factors.

Q. As the common multiple is a product consisting of two or more factors, it follows that it may be divided by each of these factors without a remainder; how, then, may it be determined, whether one number is a common multiple of two or more numbers, or not?

A. It is a common multiple of these numbers, when it can be divided by each without a remainder.

Q. What is the common multiple of 2, 3, and 4, then?

A. 24.

Q. Why?

A. Because 24 can be divided by 2, 3, and 4, without a remainder.

Q. We can divide 12, also, by 2, 3, and 4, without a remainder; what, then, is the least number, that can be divided by 2 or more numbers, called?