To reduce an improper fraction to its equivalent whole or mixed number.

RULE.* Divide the numerato by the denominator, and the quotient will be the whole or mixed number sought.

To reduce a whole number to an equivalent fraction, having a given

denominator.

RULE.+-Multiply the whole number by the given denominator, then set the product over the said denominator, and it will form the fraction required.

EXAMPLES.

1. Reduce 9 to a fraction whose denominator shall be 7. Here 9 X 763, then 3 is the answer.

For 636379, the proof.

2. Reduce 13 to a fraction whose denominator shall be 12. 3. Reduce 27 to a fraction whose denominator shall be 11.

* This rule is evidently the reverse of the former; and the reason of it is manifest from the nature of Common Division.

+ Multiplication and Division being here equally used, the result must be the same as the quantity first proposed.

CASE V.

To reduce a compound fraction to an equivalent simple one.

RULE.*—Multiply all the numerators together for a numerator, and all the denominators together for the denominator, and they will form the simple fraction sought.

When part of the compound fraction is a whole or mixed number, it must first be reduced to a fraction by one of the former cases.

And, when it can be done, any two terms of the fraction may be divided by the same number, and the quotients used instead of them. Or when there are terms that are common, they may be omitted.

EXAMPLES.

1. Reduce of of

Here

Or,

1 X 2 X 3

2 × 3 × 4

1 X 2 X 3 1

2 X 3 X 4

=

4'

by omitting the twos and threes,

2. Reduce of off to a simple fraction.

To reduce fractions of different denominators to equivalent fractions, having a common denominator.

RULE.+-Multiply each numerator into all the denominators except its own for the new numerators; and multiply all the denominators together for a common denominator.

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 be reduced, by their proper rules, to the form of simple fractions. * The truth of this rule may be shown as follows: Let the compound fraction be of . Now of is $3, which is; consequently of will be × 2 or 1; that is the numerators are multiplied together, and also the denominators, as in the rule.-When the compound fraction consists of more than two single ones; having first reduced two of them as above, then the resulting fraction and a third will be the same as a compound fraction of two parts; and so on to the last of all.

This is evidently no more than multiplying each numerator and its denominator by the same quantity, and consequently the value of the fraction is not altered.

EXAMPLES.

1. Reduce, and to a common denominator.

1 x 3 x 4 = 12 the new numerator for 4.
2 X 2 X 4 = 16 ............................ ditto for .
3 x 2 x 3 = 18

ditto for

2 X 3 X 4 = 24 the common denominator.

Therefore the equivalent fractions are 14, 14, and 14.

Or the whole operation of multiplying may be very well performed mentally, and only set down the results and given fractions thus: 1, 3, 1, 14, 14, 11 = 12, 11, 12, by abbreviation.

6

8

2. Reduce and to fractions of a common denominator. 3. Reduce,,, to a common denominator.

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

=

35

Ans. 1,3. Ans. 40, 80, 8.

63 36 45

Ans. 8, 38, 120.

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.

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

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

CASE VII.

To find the value of a fraction in parts of the integer.

RULE.-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.*

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.

To reduce a fraction from one denomination to another.

RULE.* 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, or the denominator, if to a greater.

4. Reduce of a farthing to the fraction of a pound.

5. Reduce

cwt. to the fraction of a lb.

6. Reduce dwt. to the fraction of a lb. troy.

7. Reduce § of a crown to the fraction of a guinea.
8. Reduce § of a half-crown to the fraction of a shilling.

ADDITION OF VULGAR FRACTIONS.

To add fractions together that have a common denominator.

RULE.-Add all the numerators together, and place the sum over the common denominator, and that will be the sum of the fractions required.

If the fractions proposed have not a common denominator, they must be reduced to one. Also compound fractions must be reduced to simple ones; and mixed numbers to improper fractions; also fractions of different denominations to those of the same denomination.†

* 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 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.

EXAMPLES.

1. To add and together.

Here+= 13, the answer.

2. To add and together.

3 + 1 = 1 ÷ + 3 5 = 18 = 18 the answer.

3. To add § and 7 and 3 of together.

§ + 7} + } of } = ÷ + ! + 1 =1+ ? + 1 = 8, the answer.

4. To add

5. To add

and together.

and

6. Add and together

7. What is the sum of 8. What is the sum of 9. What is the sum of 10. What is the sum of

Ans. 14. Ans. 1. Ans.

Ans. 1183.

together.

and

and?

and
and
of a pound and

and 24?

Ans. 34%

of, and 9?

Ans. 10

of a shilling?

Ans. 13's. or 13s. 10d. 23q.

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

6. What is the difference between 53 and 4 of 4? 7. What is the difference between of a pound, and 3 of 3 of a shilling?

Ans. s. or 10s. 7d. 13q.

8. What is the difference between 2 of 5 of a pound, and of a shilling? Ans. 20377. or 11. 8s. 119.

2100

MULTIPLICATION OF VULGAR FRACTIONS.

RULE.*-Reduce mixed numbers, if there be any, to equivalent fractions;

* 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 compound fraction; which is resolved by multiplying together the numerators and the denominators.