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3. Reduce,, and to a common denominator.

36 45

Ans.,,.

4. Reduce §, 23, and 4 to a common denominator. Ans. 5, 8, 30.

309

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

5

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Ex. and and, by multiplying the former by 7 and the latter by 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

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.

16

and 19. and 24.

Ex. and and 4 = 3 and and 7 = 14 and 14

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CASE VII.

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*.

* 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.

EXAMPLES.

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3. Find the value of 3 of a pound sterling.

Ans. 7s 6d.

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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 denomi nator, if to a greater.

EXAMPLES.

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

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3 × 20 × 12 = 430 160, the Answer.

=

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

2. Reduce

2. Reduce

Ans. 2d.

of a penny to the fraction of a pound. 4 × 72 × 20=, the Answer. 3. Reduce to the fraction of a penny. 4. Reduce q 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.

Ans. 1400 Ans. 32.

Ans. zo.

Ans..

7. Reduce crown to the fraction of a guinea.
8. Reduce half-crown to the fract. of a shilling. Ans.
9. Reduce 2s 6d to the fraction of a £.

10. Reduce 17s 7d 33g to the fraction of a £.

Ans..

ADDITION OF VULGAR FRACTIONS.

Ir 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 required.

Then

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

* 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.

EXAMPLES.

1. To add and together.

Here+= 1, the Answer.

2. To add and together.

S

÷ + 3 = 3 8 + 38 = 43 = 18, the Answer.

3. To add and 74 and of

4. To add and together.

together,

§ + 71⁄2 + } of ÷ =}+&{+}={+&+}=}=8}.

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and and 24?

and ÷ of, and 9? Ans. 10. pound and § of a shilling? Ans. 1's or 13s 10d 24q.

of a

11. What is the sum of 3 of a shilling and

Ans.

of a penny?

d or 7d 1139.

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

and of a penny?

3139

Ans. 1s or 3s ld 11q.

1

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 §.

27

4-4=33-286, the Answer.

3. What

1

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6. What is the diff. between 5 and 4 of 44? Ans. 4. 7. What is the difference between & of a pound, and of

of a shilling?

191

333

Ans. 1 or 10s 7d 1-3q.

8. What is the difference between

and of a shilling?

of 5 of a pound,

Ans. 337 or 1/ 8s 11-3d.

MULTIPLICATION OF VULGAR FRACTIONS.

* REDUCE mixed numbers, if there be any, to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required.

EXAMPLES.

1. Required the product of 4 and 3.

Here, the Answer.

Or × × 1 = 1/
& 3

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2

2. Required the continued product of 3, 34, 5, and 2 of 3.

7 13 5 3 3 13 × 3 Here X X X X 3 1 4 5

4.

3. Required the product of

4. Required the product of

39

=

= 47, Ans.

4 x 2

8

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5. Required the product of, 4, and 14.

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Ans.
Ans..

* 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.

Note, A Fraction is best multiplied by an integer, by dividing the denominator by it; but if it will not exactly divide, then multiply the numerator by it.

6. Required

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