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2 X 3 X 4 = 24 the common denominator.

Therefore the equivalent fractions are 1, and 14.

16

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

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. 48, 45

40

36

Ans. 5, 38,

78

120

30.

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.*

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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.

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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.

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

Ans. 32d.

Ans. 1440

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

Ans. 12.

ADDITION OF VULGAR FRACTIONS.

To add fractions together that have a common denominator.

RULE.-Add all the numerators together, and place the sum oyer 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?= 1%, the answer.

2. To add

3 + 3/

5

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= 1; + 35 = =143 the answer.

3. To add & and 71⁄2 and 1⁄2 of † together.

= { +1/+1={+%+} = % = 83, the auswer.

§ + 7 + } of 4. To add & and § together.

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Ans. 14. Ans. 1. Ans. 14

Ans. 1183.

Ans. 38.

Ans. 10

Ans. 135s. or 13s. 10d. 24q.

of a penny? Ans. 1 d. or 7d. 1434.

12. What is the sum of 4 of a pound, and 3 of a shilling, and of a penny? Ans. s. or 3s. 1d. 1199.

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SUBTRACTION OF VULGAR FRACTIONS.

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

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6. What is the difference between 53 and 4 of 4?

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Ans. 418

7. What is the difference between § of a pound, and § of † of a shilling?

Ans. s. or 10s. 7d. 1q. of a shilling? 37. or 11. 8s. 119.

8. What is the difference between 4 of 5 of a pound, and

Ans.

MULTIPLICATION OF VULGAR FRACTIONS.

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

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* 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.

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 and .

Here, X = 3/8 = , the answer.

Or, † × } = { × } = 18

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

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RULE.*—Prepare the fractions as before in Multiplication; then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide; but if not, then invert the terms of the divisor, and multiply the dividend by it, as in Multiplication.

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2. Divide by s

Here × 1 = × 254, by the latter way.

=

Ans. 4.

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Ans. 12

4. Divide by 3.

Ans. 13.

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Ans. 18

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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.

• Division being the reverse of Multiplication, the reason of the Rule is evident.

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

RULE OF THREE IN VULGAR FRACTIONS,

RULE.*—Make the necessary preparations as before directed; then multiply continually together, the second and third terms, and the first with its terms inverted as in Division, for the answer.

EXAMPLES.

1. If of a yard of velvet cost of a pound sterling; what will of a yard cost?

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2. What will 33 oz. of silver cost, at 6s. 4d. an ounce ? Ans. 11. 1s. 44d. 3. If of a ship be worth 2737. 2s. 6d., what is 1⁄2 of her worth ?

Ans. 2277. 12s. ld.

4. What is the purchase of 12302. bank-stock, at 108§ per cent?

Ans. 13361. 1s. 9d.

5. What is the interest of 2731. 15s. for a year, at 34 per cent?

Ans. 81. 17s. 114d.

6. If of a ship be worth 731. Is. 3d., what part of her is worth 250l. 10s. ? Ans..

7. What length must be cut off a board, that is 72 inches broad, to contain a square foot, or as much as another piece of 12 inches long and 12 broad?

Ans. 18 inches. S. What quantity of shalloon, that is of a yard wide, will line 9 yards of cloth, that is 2 yards wide? Ans. 31 yds. 9. If the penny-loaf weigh 6% oz. when the price of wheat is 5s. the bushel; what ought it to weigh when the wheat is at 8s. 6d. the bushel ? Ans. 4 07. 10. How much in length, of a piece of land that is 11 poles broad, will make an acre of land, or as much as 40 poles in length and 4 in breadth ?

Ans. 13 poles. travelling 13§ hours travelling only 11 Ans. 40

days.

11. If a courier perform a certain journey in 351⁄2 days, a-day; how long would he be in performing the same, hours a-day? 12. A regiment of soldiers, consisting of 976 men, are to be new clothed, each coat to contain 2 yards of cloth that is 1ğ yard wide, and lined with shalloon yard wide; how many yards of shalloon will line them? Ans. 4531 yds. I qr. 29 nails.

DECIMAL FRACTIONS.

A DECIMAL FRACTION is that which has for its denominator, a unit (1) with as many ciphers annexed as the numerator has places; and it is usually expressed by setting down the numerator only, with a point before it on the left hand.

* This is only multiplying the second and third terms together, and dividing the product by the first, as in the Rule of Three in whole numbers.

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