Q. When is one number said to be a common multiple of two or more numbers? Of what numbers is 6 a common multiple? Of what numbers is 8 a common multiple? What is the least common multiple of two or more numbers? What is the difference between a common multiple, and the least common multiple? Give the rule for finding the least common multiple. If the numbers have no common divisor what is the least common multiple?

OPERATION.

3)3.

8

9

...

3

1

8

2. Find the least common multiple of 3, 8, and 9. We arrange the numbers in a line and see that 3 will divide two of them. We then write down the quotients 1, and 3, and also the 8 which cannot be divided. Then as there is no common divisor between any two of the numbers, 1, 8, and 3, it follows that their product, multiplied by the divisor 3, will give the least common multiple sought.

1 x 8 x 3 x3=72.

3. Find the least common multiple of 6, 7, 8 and 10.

Ans. 840.

4. Find the least common multiple of 21 and 49.

5. Find the least common multiple of 2, 7,

6. Find the least common multiple of 4, 14,

Ans. 147. 5, 6 and 8. Ans. 840. 28 and 98. Ans. 196.

7. Find the least common multiple of 13 and 6.

Ans. 78.

8. Find the least common multiple of 12, 4 and 7.

Ans. 84.

9. Find the least common multiple of 6, 9, 4, 14 and 16. Ans. 1008.

10. Find the least common multiple of 13, 12 and 4.

Ans. 156.

11. What is the least common multiple of 11, 17, 19, 21 and 7? Ans. 74613.

REDUCTION OF VULGAR FRACTIONS.

§88. Reduction of Vulgar Fractions is the method of changing their forms without altering their value.

A fraction is said to be in its lowest terms, when there

is no number greater than 1 that will divide the numerator and denominator without a remainder.

Q. What is reduction? When is a fraction said to be in its lowest terms? Is one-half in its lowest terms? Is two-fourths? Is threefourths?

CASE I.

§ 89. To reduce an improper fraction to its equivalent whole or mixed number.

RULE.

Divide the numerator by the denominator, the quotient will be the whole number; and the remainder, if there be one, placed over the given denominator will form the fractional part.

EXAMPLES.

1. Reduce 4 and 67 to their equivalent whole or mixed numbers.

OPERATION.

4)84

Ans. 21

OPERATION.

9)67 Ans. 7층

It was shown in § 44, that the value of every fraction is equal to the quotient arising from dividing the numerator by the denominator: hence the value of the fraction is not changed by the reduction.

Q. How do you reduce a fraction to its equivalent whole or mixed number? Does this reduction alter its value? Why not? What is four-halves equal to ? Eight-fourths? Sixteen-eighths? Twenty-fifths? Thirty-six-sixths? Four-thirds? What is nine-fourths equal to? Four-fifths? Seventeen-sixths? Eighteen-sevenths?

2. Reduce to a whole or mixed number.
3. In of yards of cloth, how many yards?

Ans. 123.

Ans. 2 yd.
Ans. 5 bu.

4. In 1 of bushels, how many bushels? 5. If I give of an apple to each one of 15 children, how many apples do I give?

Ans. 5.

6. Reduce 2, 153, 6941, 72301' 327 3672 50287 987625, to their whole or mixed numbers.

7..If I distribute 878 quarter apples among a number of boys, how many whole apples do I use?

Ans: 2192.

CASE II.

§ 90. To reduce a mixed number to its equivalent improper fraction.

RULE.

Multiply the whole number by the denominator of the fra:tion; to the product add the numerator, and place the sum over the given denominator.

EXAMPLES.

1. Reduce 4 to its equivalent improper fraction. Here 4x5 20: then 20+4=24; hence,

24 is the equivalent fraction.

Ans. 2.

This rule is the reverse of Case I. In the example 4 we have the integer number 4 and the fraction . Now 1 whole thing is equal to 5 fifths, and 4 whole things are equal to 20 fifths; to which, add the 4 fifths, and we obtain the 24 fifths.

Q. How do you reduce a mixed number to its equivalent improper fraction? How many fourths are there in one? In two? In three? How many sixths in four and one-sixth? In eight and two-sixths? In seven and three-sixths? In nine and five-sixths? In ten and fivesixths? How many eighths in two and one-eighth? In three and three-eighths? In four and four-eighths? In five and six-eighths? In seven and seven-eighths? In eight and seven-eighths?

2. Reduce 47 to its equivalent improper fraction?

Ans. 287. 3. Reduce 67631, 87433, 690476, 367184, to their equivalent improper fractions.

10

Ans. 34513, 7899, 69047, 38177

1876

51

100

104

4. Reduce 847365, 874874, 67426398, to their equiva

lent improper fractions.

5. How many 200ths in 675187

6. How many 151ths in 18741

CASE III.

Ans. 135187.

Ans. 28278.

§ 91. To reduce a fraction to its lowest terms.

RULE.

I. Divide the numerator and denominator by any number that will divide them both without a remainder, and then divide

the quotients arising in the same way until there is no number greater than 1 that will divide them without a remainder.

II. Or, find the greatest common divisor of the numerator and denominator and divide them by it. The value of the fraction will not be altered by the reduction.

EXAMPLES.

1. Reduce 70 to its lowest terms.

5) 70 7)14 2

=

5)175-7)355',

1st METHOD.

which are the lowest terms of the

fraction, since no number greater than 1 will divide the numerator and denominator without a remainder.

2nd METHOD, BY THE COMMON DIVISOR.

Q. When is a fraction in its lowest terms? (see § 88.] How do you reduce a fraction to its lowest terms by the first method? By the second? What are the lowest terms of two-fourths? Of six-eighths? Of nine-twelfths? Of sixteen-thirty-sixths? Of ten-twentieths? Of fifteen-twenty-fourths? Of sixteen-eighteenths? Of nine-eighteenths?

7. Reduce to its lowest terms by the 2nd method.

Ans..

8. Reduce 315 to its lowest terms by the 2nd method.

9. Reduce

Ans. 57 to its lowest terms by the 2nd method. Ans. 13.

623

792

10. Reduce 19 to its lowest terms by the 2nd method.

Ans.

CASE IV.

§ 92. To reduce a whole number to an equivalent fraction having a given denominator.

RULE.

Multiply the whole number by the given denominator, and set the product over the said denominator.

EXAMPLES.

1. Reduce 6 to a fraction whose denominator shall be 4. Here 6×4=24; therefore 24 is the required fraction. It is plain that the fraction will in all cases be equal to the whole number, since it may be reduced to the whole number by Case I.

Q. How do you reduce a whole number to an equivalent fraction having a given denominator? How many thirds in 1? In 2? In 3? In 4? If the denominator be 5, what fraction will you form of 5? Of 4? Of 9? Of 7? Of 8? With the denominator 6, what fraction will you form of 3? Of 4? Of 5? Of 6? Of7? Of 9?

2. Reduce 15 to a fraction whose denominator shall be 9. Ans. 135. 3. Reduce 139 to a fraction whose denominator shall be 175.

Ans. 24325

175

4. Reduce 1837 to a fraction whose denominator shall shall be 181.

5. If the denominator be 837, what fractions will be formed from 327? From 889? From 575 ?

CASE V.

93. To reduce a compound fraction to its equivalent simple one.

EXAMPLES.

1. Let us take the fraction 2 of 5.

First, 3×1: hence the fractions may be written of 3×1 of; that is, three times one fourth of But of: hence we have,

a result which is obtained by multiplying together the numerators and denominators of the given fractions.