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CASE III.

To Reduce an Improper Fraction to its Equivalent Whole or

Mixed Number.

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

EXAMPLES.

1. Reduce to its equivalent number.

Here or 12 + 3 = 4, the Answer. 2. Reduce to its equivalent number.

Here » or 15 • 7 = 24, the Answer. 3. Reduce to its equivalent number. Thus, 17) 749 ( 4477

68

69
68

So that 24 = 441', the Answer.

4. Reduce to its equivalent number.
5. Reduce 'to its equivalent number.
6. Reduce ?948 to its equivalent number.

Ans. 8.
Ans. 5415.
Ans. 17145.

CASE IV.

To Reduce a Whole Number to an Equivalent Fraction, having

a Given Denominator. + MULTIPLY the whole number by the given denominators then set the product over the said denominator, and it will form the fraction required.

* 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.

EXAMPLES

EXAMPLES.

1. Reduce 9 to a fraction whose denominator shall be 7.

Here 9 x1 = 63: then is the Answer;

For 3 = 63 +7= 9, the Proof. 2. Reduce 12 to a fraction whose denominator shall be 13.

Ans. 3. Reduce 27 to a fraction whose denominator shall be 11.

I56

Ans. 297.

CASE V.

To Reduce a Compound Fraction to an Equivalent Simple One.

* MULTIPLY all the numerators together for a numerator, and all the denominators together for a 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, or cancelled.

EXAMPLES.

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1. Reduce of off to a simple fraction.

1x 2 x 3 6 1 Here

the Answer.
2 X 3 X 4 24
1x1 x3

by cancelling the 2's and 3's.
1 x 3 x 4

Or,

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* The truth of this Rule may be shown as follows : Let the compound fraction be of. Now of į is +3, which is consequently of will be X 2 or ; 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 faction and a third will be the saine as a compound fraction of two parts; and so on to the last of all,

2. Reduce

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2. Reduce jof} of H to a simple fraction.

2 x 3 x 10 60 12 4 Here

the Answer. 3 X5 XII 165 33 11

2 x 3 x ro 4 Or,

the same as before, by cancelling 3x5x11 the 3's, and dividing by 5's. S. Reduce of to a simple fraction.

Ans. 12 4. Reduce of of to a simple fraction. 5. Reduce of şof 3 to a simple fraction.

Ans. 6. Reduce şof of of 4 to a simple fraction. Ans. {. 7. Reduce 2 and į of {to a fraction.

. Ans. š.

Ans.

CASE VI.

To Reduce Fractions of Different Denominators, to Equivalent

Fractions having a Common Denominator. MULTIPLY each numerator by 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 first be reduced, by their proper Rules, to the form of simple fractions.

EXAMPLES.

1. Reduce s, š, and , to a common denominator.

1 x 3 x 4 = 12 the new numerator for 1
2 x 2 x 4 = 16

ditto
3 x 2 x 3 = 18

ditto 2 x 3 x 4 = 24 the common denominator. Therefore the equivalent fractions are in t, and

Or the whole operation of multiplying may be best per-
formed mentally, only setting down the results and given
fractions thus: 1, 3=1, 14, = r, s, ts, bý
abbreviation.
2. Reduce ? and to fractions of a common denominator.

Ans. 19
,

35
63

* 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.

3. Reduce

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

Ans. 48, 48, 44. 4. Reduce , 2, and 4 to a common denominator.

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120

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Note 1. When the denominators of two given fractions have a common measure, let them te divided by it; then multiply the terms of each given fraction by the quotient arising from the other's denominator. Ex. , and =145 and 195, by multiplying the former 5

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. 1 and = and fa, by mult. the former by 2.

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 conimon denominator.

Ex. į and } and } = { and and } = 1 and 1 and 14.

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 denoininator 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.

EXAMPLES.

1. What is the 4 of 21 6s?
f

2.What is the value of of 112 By the former part of the Rule, By the 2d part of the Rule, 21 6s

2 4

20

5) 9 4 Ans.

11 165, 9d 239

3) 40 (13s 4d Ans.

1 12

3) 12 ( 4d

3. Find the value of of a pound sterling.

Ans. 7s 6d. 4. What is the value of of a guinea ? Ans. 4s 8d. 5. What is the value of of a half crown? Ans. Is 10 d. 6. What is the value of of 4s 10d? Ans. Is Ild. 7. What is the value of 4 lb troy? Ans. 9 oz 12 dwts. 8. What is the value of is of a cwt ? Ans. I

qr 7 lb. 9. What is the value of of an acre

cre? Ans. 3 ro. 20 po. 10. What is the value of is of a day? Ans. 7 hrs 12 min.

CASE VIII.

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.

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1. Reduce of a pound to the fraction of a penny. 3 x 40 x 1

160, the Answer.

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

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2. Reduce

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