80. To reduce a mixed number to an improper fraction.

RULE.-Multiply the whole number by the denominator of the fraction, and to the product add its numerator; under this sum write the denominator of the fraction.

Ex. 1. Reduce 43 to an improper fraction?

This rule is the reverse of the preceding. The reason is the

Any number may be written in a fractional form by putting 1 under it for a denominator.

81. We may give a whole number any denominator, if we first multiply it by that denominator.

For, 8-8x=48, and 10-50; also, 20=20Xg=10o.

1. Reduce 19, 10, 30, to fractions whose denominators shall be 11, 17, 29, respectively?

Ans. 2, 7, 870.

2. Reduce 27, 39, 165, to fractions whose denominators shall be 39, 46, 57, respectively? Ans. 1883, 184, 9395.

46

82. We may also reduce a fraction to another fraction having any denominator, if the denominator of such a fraction be a factor of the number to which it is to be reduced.

Thus, can be brought to a fraction whose denominator is

16; since 16=4x4, we have merely to multiply the terms of

the fraction by 4, and we have

1X4 4x4

We cannot, however, reduce to a fraction whose denominator shall be 18, since 4 is not one of the factors of 18.

We may reduce and to equivalent fractions, having 48 for denominator. For, 24 (multiplying the terms by 24), and = (multiplying the terms by 6).

THE GREATEST COMMON MEASURE.

83. If one number divides another without a remainder, it is said to be a MEASURE of that other.

A measure is another name for factor.

If any number be a measure or factor of two or more others, it is called a common measure of them; that is, a measure common to them. The numbers 12 and 18 have each the following measures:-12 has 12, 6, 4, 3, 2; 18 has 18, 9, 6, 3, 2. Therefore 12 and 18 have 6, 3, 2, common measures. 6 is their greatest common measure, that is, the greatest measure common to them.

84. To find the greatest common measure of any two numbers.

RULE.-10. Divide the greater number by the less; divide the less by the remainder; continue the process, always dividing the preceding divisor by the remainder, until there is no remainder.

2o. The last divisor will be the greatest common measure (see note 22)..

Ex. 1. Find the greatest common measure of 12 and 18?

We divide by 12, and get 6 for remainder; we then divide 12, the first divisor, by 6, and find no remainder: therefore 6 is the greatest common measure of 12 and 18.

12)18(1

12

6)12(2

Find the greatest common measure of

1. 48 and 72, 66 and 154, 195 and 819

2. 246 and 1084, 378 and 1467, 18996 and 29984

3. 128 and 7364, 365 and 7345, 1964 and 78688

on.

For more than two numbers.

ANS. 24, 22, 39

2, 9, 4

4, 5, 4

RULE. Find for two then of this measure and a third, and so

4. 12, 42, and 66; 19, 57, and 380

ANS.

6, 19

6. 14, 70, and 1484; 8128, 14816, 75288, and 8472

36, 8

5. 324, 612, and 1032; 96, 216, 88, and 16

85. To reduce fractions to their lowest terms.

14, 8

RULE I.-Divide the terms of the fraction by any common factors; the new terms in the same manner, and so on, until you get terms having no common factors.

RULE II.-Divide the terms of the fraction by their greatest

common measure.

Ex. 1. Reduce to its lowest terms?

Here, 12-5-3 Ans. This result is found by dividing by 2 continually. But we might have divided by 4, &c.

The greatest common measure of 48 and 112, we find (84) is 16, then dividing the terms by 16, we get 4, as before.

The following remarks will be found useful in finding common factors:

1. All even numbers are divisible by 2 at least.

2. All even numbers are divisible by 4, when the number expressed by their last two figures is divisible by 4.

3. A number is divisible by 8, if its units' period be divisible by 8.

4. An equal number of cyphers cancel when numbers end in one or more cyphers.

5. Numbers ending in 0 or 5, are divisible by 5, and no others.

6. If the sum of all the figures be divisible by 3 or 9, the number is divisible by 3 or 9.

7. A number is divisible by 11, if the sums of the alternate

figures be equal, beginning at either right or left; or, if one sum exceed the other by 11, or 22, 33, &c.

REASON.-1. This is self-evident.

2. Since 100 is divisible by 4, any number of hundreds is divisible by 4 (note 23). But any number may be considered as a certain number of hundreds the number expressed by the figures in its units' and tens' places. Therefore if these last be divisible by 4, the entire number must be divisible by 4 (note 22). Thus 1728-1700+28; 736-700+36.

3. The reason of this is similar to that of the preceding, only say 1000, instead of 100, and 8 instead of 4.

1.

4. This is obvious.

5. Go over the MULTIPLICATION TABLE by 5.

Reduce to their lowest terms:

168, 2. 638 138

146,

9,8, 1896、

1396

9687 T12

ANS.

THE LEAST COMMON MULTIPLE.

86. When a number is a factor of another number, it has been called a measure of that other.

Any number is called a MULTIPLE of any of its measures or factors. Thus 12 is a multiple of 6, and of 4.

When a number is a multiple of two or more others, as 12 of 6, 4, and 3, it is called their Common Multiple.

The numbers 6, 4, and 3, have also 24, 36, 48, &c., common multiples. For simplicity we always endea vour to use the least common multiple of numbers.

87. To find the Least Common Multiple of two or more numbers.

RULE.-10. Place the numbers in a line; then divide by a measure or divisor, found by inspection, which will divide the largest number of the given numbers.

2o. Set down the quotients and undivided numbers.

3o. Proceed in the same manner with the quotients and undivided numbers, until no number can be found to measure any two of them.

4o. Then multiply together the undivided numbers and the divisors used; the product will be the least common multiple.

When numbers are all prime, that is, when they have no common factors, their product is their least common multiple.

Ex. 1. Find the least common multiple of 6, 12, 9, 18 24, and 30?

By looking at the numbers, we can see at once that 3 will measure more of them than any other common factor; we divide by 3, and write the quotients under the dividends. We next divide by 2; and these quotients again by 2; these again by 4; until we find 2 and 5 -numbers which have no common factors. The product of quotients

is, 3×2×2×3=36; and this 36×2×5, the undivided numbers, gives 360, the required least common multiple.

REASON. This process is merely using one of the factors common to several numbers instead of all the factors the same as itself: therefore, as only prime and common factors are used the product is the least possible; and since no factors have been omitted, the product must be a multiple of all the numbers.

88. To reduce fractions to equivalent ones having a Common Denominator.

RULE. Find the least common multiple of all the denomi. nators, which will be the required common denominator; then proceed as in 82.

This is merely an extension of 82, and we proceed on the same principle.

REASON. We have seen that all the denominators must measure that denominator to which they can be reduced: hence the reason of the rule.