Note, It is evident, that in this and several other operations, when any of the proposed quantities are integers, or mixed numbers, or compound fractions, they must first be reduced, by their proper Rules, to the form of simple fractions.

EXAMPLES.

1. Reduce, 3, and 4, to a common denominator. 1 x 3 x 4 = 12 the new numerator for

2 X 3 X 4 = 24 the common denominator. Therefore the equivalent fractions are 1, 4, and 1.

Or the whole operation of multiplying may often be performed mentally, only setting down the results and given fractions thus,, 3, 1, = 11, 14, 14, 15, 1, by abbre.

viation.

2. Reduce and to fractions of a common denominator. Ans.,.

3. Reduce }, }, and to a common denominator.

45

Ans. 8, 8, 18.

4. Reduce, 23, and 4 to a common denominator.

Ans.,,.

Note 1. When the denominators of two given fractions have a common measure, let them be divided by it; then multiply the terms of each given fraction by the quotient arising from the other's denominator.

Ex. 1 and 3 = 14'; and, by multiplying the former by 7 and the latter by 5.

5

2. When the less denominator of two fractions exactly divides the greater, multiply the terms of that which has the less denominator by the quotient.

Ex. ‡ and 1⁄2 = ¦ and †, by mult. the former by 2.

3. When more than two fractions are proposed, it is sometimes convenient, first to reduce two of them to a common denominator; then these and a third; and so on till they be all reduced to their least common denominator.

Ex.

=

and and 7 = and ÷ and 1 and 1 and .

CASE VII.

To reduce Complex Fractions to single ones.

REDUCE the two parts both to simple fractions; then mul. tiply the numerator of each by the denominator of the other; which is in fact only increasing each part by equal multi

plications, which makes no difference in the value of the

To find the value of a Fraction in Parts of the Integer.

MULTIPLY the integer by the numerator, and divide the product by the denominator, by Compound Multiplication and Division, if the integer be a compound quantity.

Or, if it be a single integer, multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator. Then, if any thing remains, multiply it by the parts in the next inferior denomination, and divide by the denominator, as before; and so on as far as necessary; so shall the quotients, placed in order, be the value of the fraction required.*

EXAMPLES.

1. What is the of 21 6s? 2. What is the value of of 11? By the former part of the Rule By the 2d part of the Rule,

3. Find the value of

3) 12 (4d

Ans. 7s 6d.

of a pound sterling. 4. What is the value of of a guinea? 3

5. What is the value of 2 of a half crown?

Ans. 4s 8d. Ans 1s 10d.

Ans. 1s 11 d. Ans. 9 oz 12 dwts.

Ans. 1 qr 7 lb.

Ans. 3 ro 20 po.

Ans. 7 hrs 12 min.

The numerator of a fraction being considered as a remainder, in Division, and the denominator as the divisor, this rule is of the same nature as Compound Division, or the valuation of remainders in the Rule of Three, before explained.

CASE IX.

To Reduce a Fraction from one Denomination to another.

* CONSIDER how many of the less denomination make one of the greater; then multiply the numerator by that number, if the reduction be to a less name, but multiply the denominator, if to a greater.

EXAMPLES.

1. Reduce of a pound to the fraction of a penny. 1 × 3 × 1 = 43° = 180, the Answer.

0

2. Reduce of a penny to the fraction of a pound.

7. Reduce

crown to the fraction of a guinea.

8. Reduce half-crown to the fract. of a shilling. Ans. 1.

Ans..

9. Reduce 2s 6d to the fraction of a £. 10. Reduce 17s 7d 339 to the fraction of a £. Ans. Ht.

Ans.

ADDITION OF VULGAR FRACTIONS.

IF the fractions have a common denominator; add all the numerators together, then place the sum over the common denominator, and that will be the sum of the fractions re. quired.

† If the proposed fractions have not a common denominator, they must be reduced to one. Also compound fractions

*This is the same as the Rule of Reduction in whole numbers from one denomination to another.

+ Before fractions are reduced to a common denominator, they are quite dissimilar, as much as shillings and pence are, and therefore cannot be incorporated with one another, any more than these can. But when they are reduced to a common denominator, and made parts of the same thing, their sum, or difference, may then be as properly expressed by the sum or difference of the numerators, as the sum or difference of any two quantities whatever, by the sum or difference of

must be reduced to simple ones, and fractions of different denominations to those of the same denomination. Then add the numerators, as before. As to mixed numbers, they may either be reduced to improper fractions, and so added with the others; or else the fractional parts only added, and the integers united afterwards.

+张
18

43 = 113, the Answer.

3. To add and 7 and of together.

{+}+} of } = {+Y+1 = {+%° +z = Y = 8}.

4. To add and together.

Ans. 14.

Ans. 11.

Ans..

their individuals. Whence the reason of the Rule is manifest, both for Addition and Subtraction.

When several fractions are to be collected, it is commonly best first to add two of them together that most easily reduce to a common denominator; then add their sum and a third, and so on.

Note 2. Taking any two fractions whatever, and 35, for example, after reducing them to a common denominator, we judge whether they are equal or unequal, by observing whether the products 35 x 11, and 7X 55, which constitute the new numerators, are equal or unequal. If, therefore, we have two equal products 35 x 11 = 7 × 55, we may compose from them two equal fractions, as 35 = 4, or 3,5 = ff.

If, then, we take two equal fractions, such as and 3, we shall have 35 X 11 = 7 × 55; taking from each of these 7 × 11, there will

remain (357) × 11 = (55 — 11) × 7, whence we have

35- 7

55 - 11

=

were respectively added to

35+ 7
55 +11

=

, it may in a similar way be shown, that

a+b

c+d

Hence, when two fractions are of equal value, the fraction formed by taking the sum (or the difference) of their numerators respe tirely, and of their denominators respectively, is a fraction equal in value to each of the original fractions. This proposition will be found useful in the doctrine of proportions.

11: What is the sum of of a shilling and

៖

Ans.

12. What is the sum of of a pound, and

of a

13s 10d 23q.

d or 7d 113q.

of a shilling,

SUBTRACTION OF VULGAR FRACTIONS.

PREPARE the fractions the same as for Addition, when necessary; then subtract the one numerator from the other, and set the remainder over the common denominator, for the difference of the fractions sought.

EXAMPLES.

1. To find the difference between and . Here=: =, the Answer.

2. To find the difference between and §.

= , the Answer.

Ans. .

and ?

Ans. รัฐ·

31

2 { = 37 - 38 3. What is the difference between and ? 4. What is the difference between 5. What is the difference between and? Ans. 6. What is the diff. between 5 and 4 of 41 ? Ans. 4. 7. What is the difference between of a pound, and of of a shilling? Ans.'s or 10s 7d 11q. 8. What is the difference between 2 of 5 of a pound, and 3 of a shilling. Ans. 1 or 11 8s 113d.

100

MULTIPLICATION OF VULGAR FRACTIONS.

* REDUCE mixed numbers, if there be any, to equivalent

* Multiplication of any thing by a fraction, implies the taking some part or parts of the thing; it may therefore be truly expressed by a