We know (ART. 34, PROP. III.,) that the value of a fraction is not changed by multiplying both numerator and denominator by the same number. If, then, we multiply the numerator and denominator of each fraction by the product of the denominators of all the other fractions, we shall retain the values of the respective fractions, and at the same time they will have a common denominator. Let it be required to reduce,of,, and of, to equivalent fractions having a common denominator. These fractions, when reduced to their simplest form, become 1, 3, 1, and 2.

For first fraction,

Multiply the numerator and denominator, each by 3× 11x9, the product of the denominators of the other fractions, and we find

1×3×11×9-297 for new numerator.

Multiply the numerator and denominator, each by 2× 11x9, the product of the denominators of the other fractions, and we find

2×2×11x9=396 for new numerator.

3x2x11x9=594

66 66

For third fraction,

denominator.

Multiply the numerator and denominator, each by 2× 3x9, the product of the denominators of the other fractions, and we find

3 ×2×3×9=162 for new numerator.

11×2×3×9=594

For fourth fraction, 3.

"6 66 denominator.

Multiply the numerator and denominator, each by 2x

3 × 11, the product of the denominators of the other frac

tions, and we find

·2×2×3×11-132 for new numerator.

9×2×3×11=594

66 66

denominator.

Hence the fractions become 37, 394. 194, and 134.

It will be seen that each numerator is multiplied by the product of all the denominators except its own. It will also be seen that in obtaining each new denominator, the factors are the same, namely, all the denominators.

Hence the following

RULE.

Reduce mixed numbers to improper fractions, and com pound fractions to their simplest form. Then multiply each numerator by all the denominators except its own for a new numerator, and all the denominators together for a common denominator.

Repeat this Rule.

EXAMPLES.

1. Reduce,, to equivalent fractions having a common denominator. Ans. 10, TU, TOS..

2. Reduce,,, to equivalent fractions having a common denominator.

Ans. 1, 11, 1 3. Reduce,,,, to equivalent fractions having a common denominator. 48, 79, 88, 308. 4. Reduce of 3, 4, 5, to equivalent fractions having

Ans.

Ans. 16, il, †g.

a common denominator. 5. Reduce of of 5, 7, 51, to equivalent fractions having a common denominator. Ans. 75, 220, 56.

41. In most cases fractions may be reduced to equiv alent ones having a smaller common denominator than is given by the above rule. Before showing how to find the

least common denominator of fractions, it becomes necessary to show how to find

THE LEAST COMMON MULTIPLE.

A multiple of several numbers is such a number as can be divided by each of them without a remainder. Thus, 12, 24, 36, 48, &c., are multiples of 2, 3, 4, and 6, since each of them is divisible by 2, 3, 4, and 6. Any set of numbers may have an infinite number of multiples. In practice it is the least common multiple which is usually sought. In the above example, 12 is the least common multiple of 2, 3, 4, and 6.

Let us seek the least common multiple of two numbers, as for example, of 4 and 18. Separating these numbers into their smallest component parts, (ART. 21,) they become 4=2x2; 18=2×3×3. If we multiply 2×2=4 by 2×3×3=18, we shall obtain 2×2×2×3×3, which is obviously a common multiple of 4 and 18, since the factors of these numbers are found in this expression. But it is not the least common multiple of 4 and 18, since one of the 2's, which is a common factor of 4 and 18, may be omitted, and the result, 2×2×3×3, will still contain all the different factors of 4 and 18. Hence, when two numbers have no common divisor, their least common multiple may be found by taking their product. When they have a common divisor, their least common multiple may be found by dividing their product by their greatest common divisor; or, by dividing one of the numbers by their greatest common divisor, and multiplying the quotient by the other number; or, by dividing each number by their greatest common divisor, and multiplying the product of the quotients by this greatest common divisor.

It is the last method that we find most convenient to

employ.

The least common multiple of more than two numbers may be found, by first finding the least common inultiple of any two of the numbers, and then finding the least common multiple of that multiple, and another of the given numbers, and so on, until all the different numbers have been used.

We will now seek the least common multiple of 10, 18 and 21.

The greatest common divisor of 10 and 18 is 2. Dividing 10 and 18 by 2, we find 5 and 9 for the quotients; hence the least common multiple of 10 and 18 is 2×5×9. We now seek the least common multiple of 2×5×9 and 21. The greatest common divisor of these two numbers is 3, it being a divisor of 9 and of 21.

Dividing by 3, we have 2x5 x 3 and 7 for the quotients. Hence the least common multiple of 2×5x9=90 and 21, is 3×2×5×3×7=630, which is also the least common multiple of 10, 18 and 21.

If we place the numbers 10, 18 and 21 in a horizontal line, and divide the 10 and 18 by 2, and bring down the 21, we shall obtain a second line consisting of 5, 9, and 21. Dividing the 9 and 21 of this

2 10, 18, 21

3

5, 9, 21

5,

3, 7

2×3×5×3×x7=630.

second line by 3, we obtain a third line consisting of 5,3, and 7, no two of which have a common divisor. Now, multiplying the divisors 2 and 3 by the product of the numbers in the last horizontal line, we have 630, the leas". common multiple sought.

Hence the least common multiple of any set of numbers may be found by the following

RULE.

Write the numbers in a horizontal line; divide them by the least number which will divide two or more of them without a remainder; place the quotients with the undivided numbers, if any, for a second horizontal line; proceed with this second line as with the first; and so continue until there are no two numbers which can be exactly divided by the same divisor. The continued product of the divisors, and of the numbers in the last horizontal line, will give the least common multiple.

NOTE. When there is no number which will divide two of the given numbers, their continued product must be taken for the least common multiple.

What is a multiple of several numbers? Mention some of the multiples of 2, 3, 4, and 6. Are the number of multiples of any set of numbers limited? Repeat the Rule for finding the least common multiple of any set of numbers. When there is no number which will divide two of the given numbers, how is the least multiple found?

EXAMPLES.

1. What is the least common multiple of 12, 16, and 24?