therefore the least common multiple = 3 × 5 × 7 = 105; therefore the fractions become

In each of the equivalent fractions, we have unity divided into 105 equal parts, and those fractions represent respectively 49, 50, and 48 of such parts; therefore the sum of the fractions must represent 49+50+48 147 105*

or 147 such parts, that is, must be

Note 1. If the sum of the fractions be a fraction which is not in its lowest terms, reduce it to its lowest terms; and if the result be an improper fraction, then reduce it to a whole or mixed number: thus 1==1}}: the same remark applies to all results in Vulgar Fractions.

Note 2. Before applying the rule, reduce all fractions to their lowest terms, improper fractions to whole or mixed numbers, and compound fractions to simple ones.

Note 3. If any of the given numbers be whole or mixed numbers; the whole numbers may be added together as in simple addition, and the fractional parts by the Rule given above.

First, find the least common multiple of the denominators;

therefore the least common multiple

=2×5×4×3×11=1320;

(4) 217+6}+{+} of {2+}} + } of 22.
(5) 2+3+4+58 +64.

(6) 14+3+1+18+2+7 of 7.

(7) 5+ of 4 of 3+9+ of § of 4.

(8) § of 12+ of 3 +33 of 18 of 14+1 of 32 of of 1.

(9) 270+65050001+53 +1.

(10) of 2+1 of (1+18)+31+8} of {1+1}.

SUBTRACTION.

76. RULE. Reduce the fractions to their least common denominator, take the difference of the new numerators, and place the common denominator underneath.

Proceeding by the Rule given above, since 8 is clearly the least common multiple of the denominators, the equivalent fractions will be and,

The unit in each of the equivalent fractions is divided into 8 equal parts, and there are 7 and 4 parts respectively taken, and therefore the

difference must be 3 of such parts, or, in other words, the difference of the two fractions is g.

Note 1. Remember always, before applying the above Rule, to reduce fractions to their lowest terms, improper fractions to whole or mixed numbers, and compound fractions to simple ones.

Note 2. If either of the given fractions be a whole or mixed number, it is most convenient to take separately the difference of the integral parts and that of the fractional parts, and then add the two results together, as in the following examples.

Ex. 1. From 43 subtract 21.

Here 4-2=2, and 3-1-3-3=};

therefore the difference of 43 and 21=21.

Now cannot be taken from 1, since it is the greater of the two; we therefore add 1 to 1, and take & from 1+1 or ; and then, in order that the difference may not be altered, we add 1 to the 2.

Now

4-3=1;

therefore the difference of 41 and 23=13.

For the process expressed at length is

4+1-(2+)

which =4+1+1-(2+1+3) (adding and subtracting 1),

(10) 13 and 9.

(11) 50

and 47

(12)

42 and 30.

(20) of of of 82 and of 4 of 11 of 17.

2. By how much does of- of exceed & of 3 of 1? 3. Add of to 27 and subtract & from the result.

4. From the sum of 113 and 87 subtract 912.

5. By how much does the difference of 533 and 24 exceed the sum of ++?

of+

6. By how much does the sum of the fractions 1 and exceed their difference ?

MULTIPLICATION.

77. RULE. Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator.

If be multiplied by 5, the result is 45, Art. (64).

But this result must be 8 times too large, since, instead of multiplying by 5, we have only to multiply by g, which is 8 times smaller than 5, or, in other words, is one-eighth part of 5. Consequently the product above, viz. 5 must be divided by 8, and 5÷8=18, Art. (65).

Note 1. The same reasoning will apply, whatever be the number of fractions which have to be multiplied together.

Note 2. Before applying the above Rule mixed numbers must be reduced to improper fractions.

Note 3. It has been shewn that a fraction is reduced to its lowest terms by dividing its numerator and denominator by their greatest common measure, or, in other words, by the product of those factors which are common to both: hence, in all cases of multiplication of fractions, it will be well to split up the numerators and denominators as much as possible into the factors which compose them; and then, after