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DIVISION OF FRACTIONS.

46. Let us endeavor to divide by . We know that can be divided by 5, by multiplying the denominator by 5, (see PROP. II., ART. 34,) which gives

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Now, since is but one eighth of 5, it follows that divided by must give a quotient eight times as great as + divided by 5. Therefore, divided by must give

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Hence, to divide one fraction by another, we have this

RULE.

Reduce the fractions to their simplest form. Invert the divisor, and then proceed as in multiplication.

If either the dividend or divisor is a whole number, it may be converted into an improper fraction having 1 for its denominator.

Repeat the Rule for the Division of Fractions.

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7. What is the quotient of 4 divided oy 17?

Ans. .

8. What is the quotient of divided by 10? Ans.

9. Divide of 4 by of.

Ans. 1-3.

10. Divide 3 of 2 by 41.

11. Divide by of 4.

Ans. L=147.

Ans. 14.

12. Divide the sum of 2, 4, 4, %, by the sum of 1, 2, 1

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13. Divide the sum of 4, 8, 7, 4, f, 4o, tt, H, H, H, by the sum of 1, 4, 1, 1, 7, t, t, to, ti, te, ts.

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14. Divide of of of by of ‡ of 4 of §.

Ans. 133.

15. Divide the sum of 1, 11, 21, 31, by the sum of 13, 21, 31. Ans. 485-1787. 16. Divide the sum of of, of, by the sum of of , of 7.

Ans.

17 Divide off of byt of 4 of 4.

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Ans. 8.

18. Divide of of by of of 12.

Ans. T.

19. Divide of of of by of of 8.

Ans.

RECIPROCALS OF NUMBERS.

47. The reciprocal of a number is the result obtained by dividing 1 by the number. Thus, the reciprocals of 2, 3, 4, and 5, are 1,,, and . From this we discover that the reciprocal of an integer, or whole number, is equal to a vulgar fraction whose numerator is 1, and whose denominator is the given number.

The reciprocal of is found by dividing 1 by ; which (ART. 46,) is 1+3=1×4=

In the same way we find the reciprocal of to be, and in general, the reciprocal of a vulgar fraction is the value of the fraction when inverted.

NOTE. From this, we see that dividing by any number is in effect the same as multiplying by the reciprocal of that number. So that operations of division may be included under those of muitiplication. A practical application of this principle may be seen under Reduction of Denominate Fractions. (ART. 89.)

EXAMPLES.

1. What are the reciprocals of 7, 8, 9, 10, 117

Ans. t, t, t, to, th

2. What are the reciprocals of 18, 23, and 41 ?

Ans. 1, 2,

Ans. 1, 4, 4, 8.

3. What are the reciprocals of, 4, 4, 5?
4. What are the reciprocals of 1, 2, 34?
5. What are the reciprocals of of 4, 4 of 7?

Ans. †,†, t

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9. Reduce 10. Reduce 1 to a mixed number. 11. Reduce 37 to a mixed number. 12. Reduce 48 to a whole number. 13. Reduce to a mixed number.

14. Reduce 2371 to a mixed number.

907 10 1

15. Reduce 1 to a mixed number.

16. Reduce 31⁄2 to an improper fraction.

17. Reduce 15 18. Reduce 3

1

13

to an improper fraction.

to an improper fraction.

19. Reduce TT to an improper fraction.

20. Reduce 1001 to an improper fraction.

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2

9

Ans. 1TT

Ans. 7.

Ans. 8.
Ans. 37.

Ans. 2557.
Ans. 17
Ans.

Ans. 20.

Ans. 5.

Ans. 1.

Ans. 17.

Ans. 1.

21. Reduce of of to its simplest form. 22. Reduce of off to its simplest form. Ans. 23. Reduce of off of 3 to its simplest form. Ans. 1%. 24. Reduce of of of 3 to its simplest form.

Ans..

25. Reduce of 14 of of 100 to its simplest form.

Ans. 200.

Ans. 11, 11, 12

26. Reduce,,, to equivalent fractions having a com mon denominator. 27. Reduce,, 4, 4, 4, to equivalent fractions having a common denominator. Ans. 30, 20, 15, 18, 18.

28. Reduce 3, 3, 4, r, to equivalent fractions having a common denominator.

Ans. 18, 28, 10, 20.

29. Reduce,,,, to equivalent fractions having a Ans. 15, 13515, 1855; 115.

common denominator.

385

30. Reduce,, 7, 13, to equivalent fractions having

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off. What part of the whole was thus taken away?

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37. A tree 150 feet high had broken off in a storm.

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What was the length broken off?

Ans. 30 feet.

38. A and B together possess 1477 sheep, of which A and B. How many belong to each man?

owns

39. A owns

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of a ship, valued at $15422; he sells What is the value of what A has

to B of his share.

left; also, what is the value of B's part?

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40. A cotton mill is sold for $30000, of which A owns of the whole, B and C each ownofof the whole. How many dollars does each one claim?

A claims $6000). Ans. B claims $5000.

LC claims $5000.

41. A and B have a melon, of which A owns 3, and B ; C offers them one shilling, to partake equally with them of the melon, which was agreed to. How must the shilling be divided between A and B?

Ans.

A must have of it.

{ A must have to it.

42 A farmer had of his sheep in one field, ‡ in a

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