For a fuller illustration of the subject, see Art. 129.

To reduce Fractions to integers of lower denominations, and the

Multiply the numerator of the Reduce the given numbers to given fraction by that number the lowest denomination mentioned, which expresses how many of the for a numerator, and an integer next lower denomination make 1 of the denomination required to of that denomination in which the the same denomination for a defraction is given, and divide the | nominator, and they will form the product by the denominator of the fraction required. fraction. If there be a remainder, proceed as before, until it is reduced to the lowest denomination. If there be still a remainder, place it at the right of the last answer.

OBS.-Let question 2d be written upon the blackboard, in the following manner, and illustrated.

Teacher. Express by writing upon the blackboard, what part of a pound is 138.

131

40

Scholar. 138. is. of a pound, which equals 20

3

20

T. What kind of fractions are those you have written?
S. Complex.

7. Remove the denominator of the numerator, and illustrate.

S.

40 40

3 60

20

To multiply the denominator is the same as to divide

the numerator, for, to multiply the divisor is the same as to divide the dividend.

7. Remove the denominator of the fraction, and illustrate.

40

S. 3. To divide the numerator divides the fraction, for, to divide

3

OBS.-It will be recollected that it was said (Art. 60) that a mixed number is the quotient of a division, whose divisor was the denominator of the fraction. Consequently, example 10 is the quotient of a division whose divisor was 11. We have therefore only to multiply quotient and divisor together, or to reduce a mixed number to an improper fraction, and we have the fraction of a cwt., which we divide by 20, to reduce it to the fraction of a ton. Thus :

QUESTIONS.-1. Rule for reducing fractions to integers of lower denominations? 2. Rule for reducing integers of lower denominations to a fraction of a higher ?

Reduction of Vulgar Fractions to Decimal.

Art. 130.—1. Reduce to a decimal fraction.

In this example, being a proper fraction, the numerator will not contain the denominator; but, by annexing a cipher, which reduces it to tenths, we can divide by the denominator. In 1 unit there are 10 tenths, but the example is one half of a unit; therefore, one half of 10 tenths, which is 5 tenths, will be the answer. Hence the

RULE.

I. Annex a cipher, or ciphers, to the numerator, and divide by the denominator.

II. If there be a remainder, a cipher, or ciphers, may be annexed, and the process of division carried on until there be no remainder, or the quotient is sufficiently exact.

The decimal places in the quotient must be equal to the number of ciphers annexed to the numerator.

If, after division, the quotient does not contain so many, supply the deficiency by prefixing ciphers.

2. Reduce to a decimal fraction.

3. Reduce §, †, and 3, to decimal fractions.

Ans. #5.

Ans. .625, .125, .6.

4. Reduce , †,†, 1, and 1, to decimal fractions.

Ans. .875, .375, .75, .25, .5.

5. Reduce of of to a decimal fraction.

Art. 131.—To reduce a decimal fraction to a vulgar.

RULE.

Write down the given decimal, as a numerator, and for a denominator, write 1, with as many ciphers annexed as there are figures in the numerator, and then reduce the fraction to its lowest terms. (See Art. 61.)

1. Reduce .25 to a vulgar fraction.

QUESTIONS.—1. Rule for reducing a vulgar fraction to a decimal? 2. How many decimal places must there be in the quotient? 3. If the quotient does not contain a sufficient number of figures, what is to be done ?

To reduce Integers of different denominations to a Decimal Fraction of a higher denomination, and the reverse.

4 Art. 133.-2. Reduce .375

pence 2 farthings to the decimal of a shilling to integers of lower denominations.