The continued product of the denominators of several fractions will always give a common denominator, so that we may have this rule:

To reduce fractions to a common denominator, multiply each numerator by all the denominators, except its own, for a new numerator, and all the denominators together for a new denominator.

Thus,,,,,, give us respectively

48

50

But a smaller number than 720 may be used, 60 for instance. The fractions would then read 용응, 송응, 충용, 송용, and 8. For reductions necessary in business, the denominator may usually be determined at a glance, and we prefer to omit the customary rules involving a finding of the least common multiple.

When the fractions have been reduced to a common denominator, we have the following rules : For Addition, add the numerators and place the sum over the common denominator.

For Subtraction, subtract the less numerator from the greater and place the remainder over the common denominator.

Multiplication and Division of fractions are much simpler, requiring no reduction to a common denominator.

For Multiplication, multiply the numerators together for a new numerator, and the denominators together for a new denominator.

For Division, invert the divisor and proceed as in multiplication.

Let us perform all these operations on the fractions and 13.

To add them we reduce to a common denomina

The last two problems illustrate one of the most important contractions in the use of fractions. We may regard 143×43, as a single fraction with

composite terms:

Now, as it does not

55 x 52 143 × 143 alter the value of a fraction to divide numerator and denominator by the same number, we may separate these terms into their simplest factors, 5×11× 4 × 13 11 x 13 x 11 x 13 common to both terms, which is the same as dividing both terms by these factors: thus At a single operation, we have, by steps,

These are dependent upon the same principles as common fractions, but they are treated like whole numbers, regard being had only to the decimal point.

To add Decimals, place them so that the decimal points shall be in a perpendicular line, add as in whole numbers, and place the decimal point in the sum under those in the numbers added.

To subtract, place the decimal point in the subtrahend, under that in the minuend, subtract as in whole numbers and place the decimal point in the remainder under those in the numbers employed.

To multiply, proceed as in whole numbers, and in the product point off as many places as are contained in both factors.

To divide, proceed as in whole numbers, and point off as many places in the quotient as those in the dividend exceed those in the divisor. If those in the divisor exceed those in the dividend, annex as many ciphers as there are more decimal places in the divisor.

Let us perform all these operations on 1.75 and .0875.

.0875)1.75

1.75.0875.0035) .07

.0005) .01°20

10

In the last, we factor, dividing first by 25 and then by 7. Then we annex a cipher to the .01, to make it divisible by 5. The quotient is 2, and the question is where shall the point be placed? There are three places in the dividend and four in the divisor. The excess in the divisor being one, we annex one cipher, and our quotient is 20. Suppose we divide .0875 by 1.75. Here we divide directly, getting a quotient

1.75).0875(.05 of 5. The decimal places in the dividend exceed those in the

875

divisor by two, hence we point off

two places, getting .05.

REDUCTION OF COMMON TO DECIMAL FRACTIONS. It is often more convenient to use fractions in a decimal form. To reduce them we simply divide the numerator by the denominator, observing the general rule for division of decimals.

The last is an example of what is often called a Circulating Decimal-a fraction which can be only approximately expressed in decimal form. However far the division be carried, there will be a continual remainder of 2o, giving a continual