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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 I under it; thus 7 becomes ; for, since the value of a fraction (¶ 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.

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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

part of the whole did he sell? How much is

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2. At of a dollar a yard, what will of a yard of cloth

cost? How much is of ? A. 7o.

3 Multiply & of 4 by 4.

3 X 5 X 3

45

A.

8 X 7 X 7

392'

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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, 2 of of =2 of 5 = 10 = 12, Ans.

The answers to the following examples express the fraction in its

lowest terms.

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¶ 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.

12;

Q. 12 is also produced by multiplying not only 2, but 3 and 6, likewise, each by some other number; thus, 2×6=12; 3×4= 6 x 212; 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 num bers, called?

A. The least common multiple of these numbers. 2. It sometimes happens, that one number will divide several other numbers, without a remainder; as, for instance, 3 will divide 12, 18, and 24, without a remainder; when, then, several numbers can be thus divided by one number, what is the number called?

A. The common divisor of these numbers.

Q. 12, 18, and 24, may be divided, also, each by 6, even; what, then, is the greatest number called, which will divido 2 or more numbers without a remainder?

A. The greatest common divisor.*

* In ¶ XXXVII., in reducing fractions to their lowest terms, we were sometimes obliged, in order to do it, to perform several operations in dividing; but, had we only known the greatest common divisor of both terms of the fraction, we might have reduced them by simply dividing once; hence it may sometimes be convenient to have a rule

To find the greatest common divisor of two or more numbers.

1. What is the greatest common divisor of 72 and 84?

OPERATION.

72) 84 (1

72

12) 72 (

72

A. 12, common divisor.

In this example, 72 is contained in 84, 1 time, and 12 remaining; 72, then, is not a factor of 84. Again, if 12 be a factor of 72, it must also be a factor of 84; for 72+12=84. By dividing 72 by 12, we do find it to be a factor of 72, (for 72

126 with no remainder); therefore, 12 is a common factor or divisor of 72 and 84; and, as the greatest common divisor of two or more numbers never exceeds their difference, so 12, the difference between 84 and 72, must be the greatest common divisor.

Hence, the following RULE :-Divide the greater number by the less, and, if there be no remainder, the less number itself is the common divisor; but, it there be a remainder, divide the divisor by the remainder, always dividing the last divisor by the last remainder, till nothing remain: the last divisor is the divisor sought.

Note.-If there be more numbers than two, of which the greatest common divisor is to be found, find the common divisor of two of them first, and then of that common divisor and one of the other numbers, and so on.

2. Find the greatest common divisor of 144 and 132.
3. Find the greatest common divisor of 168 and 84.
4. Find the greatest common divisor of 24, 48, and 96.

A. 12.
A. 84.

A. 24.

Let us apply this rule to reducing fractions to their lowest terms.
See T XXXVII.

5. Reduce 1 to its lowest terms.

12) 12, Ans.

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In this example, by using the common divisor, 12, found in the answer to sum No. 2, we have a number that will reduce the fraction to its lowest terms, by simply dividing both terms but once.

After the same manner perform the following examples :

6. Find the common divisor of 750 and 19), also reduce 1500 to its lowest terms. A. 250, and 1.

1. What is the least common multiple of 6 and 8? OPERATION.

2)6.8

3 4

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In this example, it will be perceived that the divisor, 2, is a factor, both of 6 and 8, and that dividing 6 by 2 gives its other factor, 3 (for 6+2=3); likewise dividing the 8 by 2 gives its other factor, 4 (for 82-4); consequently, if the divisors and quotients be multiplied together, their product must contain all the factors of the numbers 6 and 8; hence this product is the common multiple of 6 and 8, and, as there is no other number greater than 1, that will divide 6 and 8, 4x 3x2=24 will be the least common multiple of 6 and 8.

Note.-When there are several numbers to be divided, should the divisor not be contained in any one number, without a remainder, it is evident, that the divisor is not a factor of that number; consequently, it may be omitted, and reserved to be divided by the next divisor.

2. What is the common multiple of 6, 3, and 4?

OPERATION.

3) 6

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3 4

2)2 1

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1 1

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2

Ans. 3 X 2 X 2 = 12.

In dividing 6, 3 and 4 by 3, I find that 3 is not contained in 4 even; therefore, I write the 4 down with the quotients, after which I divide by 2, as before. Then, the divisors and quotients multiplied together, thus, 2 x2 X3=12, Ans.

From these illustrations we derive the following

RULE.

Q. How do you proceed first to find the least common multiple of two or more numbers ?

A. Divide by any number that will divide two or more of the given numbers without a remainder, and set the quotients, together with the undivided numbers, in a line beneath.

Q. How do you proceed with this result?

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Should it be preferred to reduce fractions to their lowest terms by XXXVII., the following rules may be found serviceable :

Any number ending with an even number or cipher is divisible by 2.

Any number ending with 5 or 0 is divisible by 5; also if it end in 0, it is divisible by 10.

A. Continue dividing, as before, till there is no number greater than 1 that will divide two or more numbers without a remainder; then multiplying the divisors and numbers in the last line together, will give the least common multiple required.

More Exercises for the Slate.

3. Find the least common multiple of 4 and 16.
4. Find the least common multiple of 10 and 15.
5. Find the least common multiple of 30, 35 and 6.
6. Find the least common multiple of 27 and 51.
7. Find the least common multiple of 3, 12 and 8.
8. Find the least common multiple of 4, 12, and 20.
9. Find the least common multiple of 2,7,14 and 49.

A. 16.

A. 30. A. 210. A. 459. A. 24.

A. 60.

A. 98.

¶ XLIII. TO REDUCE FRACTIONS OF DIFFERENT DENOMINATORS TO A COMMON DENOMINATOR.

Q. When fractions have their denominators alike, they may be added, subtracted, &c. as easily as whole numbers; for example, and are; but in the course of calculations by numbers, we shall meet with fractions whose denominators are unlike; as, for instance, we cannot add, as above, and together: what, then, may be con sidered the object of reducing fractions of different denominators to a common denominator ?

A. To prepare fractions for the operations of addition, subtraction, &c. of fractions.

Q. What do you mean by a common denominator?
A. When the denominators are alike

1. Reduce and to a common denominator.
OPERATION.

Numer. 2×6=12, new numer.
Denom. 3×6=18, com. denom.
Numer. 5 X 3=15, new numer.
Denom. 6 × 3=18, com. denom.

In performing this example, we take, and multiply both its terms by the denominator of; also, we multiply both the terms

of by 3, the denominator of; and, as both the terms of each fraction are multiplied by the same number, consequently the value of the fractions is not altered; ¶ XXXVII.

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