mination the denominator, and the fraction required is de

8.

d.

q.

12 7 3 = 607; and one pound = 960;

607

therefore is the fraction sought: because the integer being

960

divided into 960 equal parts, 12. 7. 3 contains 607 such parts.

29. In the last example we were obliged to reduce the whole to farthings; and in general, if the higher denomination do not contain the lower an exact number of times, reduce them to a common denomination, and proceed as before.

Ex.

What fraction of a guinea is half a crown?

Here sixpence is the greatest common denomination, of which a guinea contains 42, and half a crown 5, therefore 5 is the fraction required.

42

Any common denomination would answer the purpose; but, if the greatest be taken, the resulting fraction is in the lowest terms.

30. To reduce a fraction to any denomination.

Find what fraction of the proposed denomination an integer of the denomination of the given fraction is, and the fraction required will be found by Art. 16.

Ex. 1. What fraction of a pound is

2

3

2

of a pound.

of a shilling?

of 1 shilling is

of an inch?

5

213

1

of

of a pound,

=

20

Ex. 2.

1 inch is

of a yard, therefore

of an inch is of

36

7

7

36

ADDITION OF FRACTIONS.

31. If fractions have a common denominator, their sum is found by taking the sum of the numerators, and subjoining the common denominator.

1 2 3

Thus +

=

5 5 5

For, if an integer be divided into five

equal parts, one of those parts, together with two parts of the same kind, must make three such parts.

32. If the fractions have not a common denominator, reduce them to a common denominator, and proceed as before.

33.

When mixed numbers are to be added, to the sum of the fractional parts, found as before, add the sum of the inte

gers.

Ex. Add together 5, 61 and of

3
+

1

-

5

2 315 140 24 479 + =

+

+

=

4 3 35 420 420 420 420

therefore the whole sum required is 5 + 6 + 1;

SUBTRACTION.

34. The difference of two fractions which have a common denominator is found by taking the difference of their numerators, and subjoining the common denominator.

divided into five equal parts, and three of those parts be taken

from four, the remainder must be one of the parts, or

1

5

35. If the fractions have not a common denominator, let them be reduced to a common denominator, and then take the difference as before.

When mixed numbers are to be subtracted, the integers may be subtracted separately, and then the fractional parts. Thus,

And if the fractional part of the mixed number to be subtracted is greater than that of the other, deduct a unit from the greater number and add it in a fractional form to the smaller fractional part; then proceed as before. Thus,

36. DEF. To multiply one fraction by another is to take such part or parts of the former as the latter expresses.

This is done by multiplying the numerators of the two fractions together for a new numerator, and the denominators for a new denominator.

5 3 15

the definition of multiplication, of or (Art. 16).

7

4 28

If there be more than two fractions to be multiplied together, a similar rule applies :-multiply all the numerators together for a new numerator, and all the denominators for a new denominator.

Compound fractions must be reduced to simple ones, and mixed numbers to improper fractions, and they may then be multiplied as before.

Hence it appears, that a fraction may be multiplied by a whole number by dividing the denominator by that number, when this division can take place.

Much trouble is frequently saved by observing what multipliers are common to the new numerator and denominator, and striking them out (Art. 12) before the multiplication is effected.

1 5

which we see at once to be, by striking out the quantity 2 × 3 × 4

common to the numerator and denominator.

Again, if in the proposed fractions, to be multiplied together, there be any numerator and any denominator which have a common measure, divide them both by that common measure, and use the resulting quotients for the purpose of forming the required product.

37. To divide one fraction by another, or to determine how often one is contained in the other, invert the numerator and denominator of the divisor, and proceed as in multiplication.

For, from the nature of division, the divisor multiplied by the quotient must produce the dividend: therefore

7

by the same quantity, and the products must be equal;

35

21

=

,

as was found by

35

20

but = 1; therefore the quotient

the rule. And the same method of proof is applicable to

all cases.

Compound fractions must be reduced to simple ones, and mixed numbers to improper fractions, before the rule can be applied.