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CASE 3d. To subtract a fraction or a mixed number from a whole number.

Take the numerator from the denominator, and place the difference as a numerator, and under it put the denominator, and subtract one from the whole number, if the less quantity be a fraction but if it is a mixed number, one must be added to the integral part thereof, the sum subtract from the whole number, and the remainder with the fractional part annexed thereto, will be the difference required.

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CASE 4th. To take mixed numbers from mixed numbers. If the fractional parts have not a common denominator. reduce them to it, when if the numerator of the fraction belonging to the greater number exceed that belonging to the less, subtract the less from the greater, under which place the common denominator, which will give the remainder of the fractional part; the integers of the given quantities subtract as in whole numbers. But if the numerator of the fraction belonging to the less number, exceed the numerator of that belonging to the greater, subtract this numerator from its. denominator and to this remainder add the numerator of fraction belonging to the greater number, which will be the numerator of the fractional part of the remainder, under which place the common denominator, add one to the integral part of the less quantity and subtract as before.

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For a demonstration of the rules for adding and subtracting frac tions, see John Walker's Philosophy of Arithmetic, pages 146 and 147.

MULTIPLICATION OF FRACTIONS.

After having reduced the given fractions according to the former rules, mixed or whole numbers to improper fractions, and compound fractions to simple ones, &c. multiply the numerators together for a new numerator, and the denominators together for a new denominator; except when a mixed number is to be multiplied by a whole number, then the fractional part may be multiplied separately, and the result annexed or added to the product of the integers, as the case may require. NOTE-A fraction is multiplied by a whole number, either by multiplying its numerator, or dividing its denominator thereby.

If, in the given quantities to be multiplied or divided, a numerator and denominator are alike, both may be wholly expunged, and where they have a common measure or divisor,“ the quotients after dividing the contrary terms thereby, will answer as well as, or better than, the original numbers.

Multiply by }

{X} = 7%

Multiply 3 by 24

+X÷=V=71

Multiply of 5 by 13 4× V=44=1}}

Tramples.
Multiply 24 by
X=+=

Multiply 7 by 63
31 X

Multiply 2 by 2

2

51

1 15

49

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40. 3+1 + 7 — 2,5 × 43 × 18 - 21's

When one or both of the fractional parts to be multiplied, are compound fractions or quantities, the continual product of the several numbers will be that required.

Examples.

Let of of of 24 be multiplied by of of 4 ××××××+ Answer.

DIVISION OF FRACTIONS.

Prepare the quantities by the foregoing rules, then invert the divisor, and proceed in every respect, as in Multiplication. NOTE.-A fraction is divided by a whole number, by an operation the converse to that of Multiplication, that is, by the multiplication of the denominator or division of the numerator by the given divisor.

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Before the learner proceeds farther, let him prove the examples in Multiplication by Division, and those of Division by Multiplication.

QUESTIONS FOR EXERCISE IN FRACTIONS.

1. What quantity is that, which added to make one?

2. What quantity is that from which if tracted, the remainder shall be ?

3. What quantity is that to which if added, the total shall be one?

+++ shall

of + be sub

of 0 of 144 be

128

QUESTIONS FOR EXERCISE IN FRACTIONS.

4. There is a quantity, which if multiplied by of of 24, will produce no more than one; what is the cube or third power of that quantity ?

5.-What quantity is that from which if of & be subtracted, and to the remainder of 14 added, the sum shall be 2? 6.-There is a quantity, which if divided by ofwill quote 917; what is the square or second power of that quantity? 7.-If of of an cwt, is 12lb how many pounds are in a hundred weight?

8. Of what number is 156 the part?
9.-What fraction of a £ is 8s. 9d?

10.-How much is the of of 2 of 30s?
11. Of what number is 18 the

part?

12.-How much is the sum of +&£+6&d?

13. A person being asked how much money he had, answered that, and of the sum would just make 100/; how much money had he.

14.-There is a number, of which its, 4 and, together with 5, will just make 25; what is the number?

15.-If 34 yards of cloth make a suit of clothes, and that 5 yards of cloth, 1 yards wide make a suit for the same person; what was the breadth of the former cloth?

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16. Take and is from of a £ sterl, and what remains? 17-If of of of my years make 17; I demand my age? 18.-Four persons have a sum of money to divide, of which the first had, the second, the third of both their shares, and the fourth had the remainder, which was just 451 3s 4d; the shares of the other three and the sum divided is required?

19. If my cattle are one-half sheep, one-fourth cows and one-sixth horses, and that I have 20 others of different descriptions; how many of each sort had I?

20.-What part of 3 is 3 of 4? 21.-What part of 5 is 4 of 7?

22.-A schoolmaster being asked how many scholars he had, answered that as many more, and as many more would just make 108; how many had he?

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23. If the sixteenth of 20,0001, was to be divided between persons; what is the share of each?

24.-A man spent of a legacy in 5 months, of the remainder in 7 months; and had 180/ left; what was the amount of the legacy?

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