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Cancelling 3 from the 15 of the numerator and from the 9 of the denominator gives 5 in place of the former and 3 in place of the latter. As no further cancellation can be made, we multiply the remaining factors toge

5

ther, which gives

1 X 5
3 X 2

The following is a convenient form of writing the work:

21.

22.

23.

24.

25.

(h.) Reduce the following to their lowest terms:

7 × 9
8 x 12'

15 x 17 x 24
51 x 48 x 30°

26.

27.

10 x 12

8 x 15

4 x 7

3 x 14'

5 x 9

6 x 15

= for the result.

1

5

$ × 1$

9 × 18

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28.

=

29.

30.

31.

32.

33.

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34.

5

6

21 x 16 x 32 x 49
24 × 28 × 35 × 36'

17 x 13 x 36 x 28
38 x 40 x 51 x 39°

10 x 11 x 12 x 9 x 8 24 × 27 × 60 × 33 × 55° 15 x 63 x 12 x 77 33 x 40 x 56 x 144

144 x 18 x 75 x 132 150 x 1728 x 96 x 38°

299 x 77 x 374 × 262 121 x 713 x 324 × 561

86. To find a Fractional Part of a Number.

1. What is of 715?

1ST SOLUTION. § of 715 equals 5 times of 715. of 715, found by dividing 715 by 6, is 119, and 5 times 119 is 5955. Hence, & of 715 5955.

2D SOLUTION.—§ of 715 of 5 times 715. 5 times 715= = of 3575 is 595g. Hence, of 715

5955, as before.

3575, and

(a.) As of 715 equals 5 times of 715, or ↓ of 5 times 715, so of any number equals 5 times of that number, or of 5 times that number; .07 of any number equals 7 times .01 of that number, or .01 of 7 times that number, etc. Hence

(b.) To get a fractional part of a number, we either divide that number by the denominator of the fraction and multiply the quotient by the numerator, or we multiply the number by the numerator of the fraction and divide the product by the denominator.

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8. ક્રુરૢ of 4284?

9. 1 of 1235? 10.

of 1235?

1556 + 259

16. 26 times 1856?
17. 37% times 7431?
18. 19 times 3886?
19. 48 times 7932?

20. What is .009 of 32.45?

1ST SOLUTION.-.009 of 32.45 equals 9 times .001 of 32.45. .001 of 32.45, found by removing the decimal point 3 places to the left (see 14), is .03245, and 9 times .03245 .29205.

25. .623 of 4.86?
26. .001 of .001?
27. 1.3* of 34.2?
28. 2.5 of 25?

=

2D SOLUTION.-.009 of 32.45 = .001 of 9 times 32.45. 9 times 32.45 = 292.05, and .001 of this, found by removing the decimal point 3 places further to the left, is .29205, as before.

(e.) What is –

21. .05 of 375?
22. .13 of 87.3?
23. .016 of 245.03?
24. .25 of .25?

29. 2.03 of 832?
30. 1.54 of .761?
31. 3.7 of .896?
32. 2.003 of 2.003?

The 1.3 should be read as an improper fraction, as should all other numbers similarly situated.

87. Reduction of Compound Fractions to Simple Ones.

(a.) To reduce compound fraction to a simple one is merely to find a fractional part of a fraction.

1. Reduce the compound fraction off to a simple fraction, i. e. find of fz.

0

SOLUTION. of 15 may be found by writing, and then making 9 a factor of the numerator and 10 a factor of the denominator (see 83, ex. 15), cancelling and reducing as in the written work below:

9

10

11 49

21

of

5

of of 1-
54

22

3. of .

4. 7 of 1.
5. 1 of 11.

6. of .

5

12

2. Reduce of 42 of 1

to a simple fraction.

SOLUTION. — Writing 15, or its equal, 27, as the number of which the fractional part is to be obtained, we find 2 of 2 by making 49 a factor of the numerator and 54 a factor of the denominator; and of this result by making 11 a factor of the numerator and 21 a factor of the denominator. The written work would take the following form:

7. 38 of 25.

8. of 1%.
9. 17 of 11.

3

5 X 9 12 X 10 4 2

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3

7

27 X 49 X H

22 X 54 21
2

2

3

(b.) These solutions show that, to reduce a compound fraction to a simple one, we first reduce the mixed numbers, if there are any, to improper fractions, then make all the numerators of the compound fraction factors of the new numerator, and all the denominators factors of the new denominator, and then cancel and reduce.

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(c.) Reduce each of the following compound fractions to simple

ones.

712

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17.7% of
18. of
19.of of of 7.

20. of of 3 of 3 of 4%.
21. of of of 13.
22.ofofofofof of 3 of 3 of 10.

of } of § of §. of of 1% of 11.

88. Reduction of Vulgar Fractions to Decimals.

(a.) To reduce a vulgar fraction to an equivalent decimal, we have only to get the part of 1 which the given fraction indicates, carrying out the work to the required number of decimal places.

(b.) Practically, the reduction is made by annexing zeroes to the numerator and dividing by the denominator.

NOTE.-In most cases it will be sufficiently exact to carry out the division to 3 or 4 decimal places, but sometimes it is necessary to earry it out much further.

14)5.0000).35718

80

100

20

2. 1%.

3. 9.

4. 11.

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5. 1. 6. 4.

7. §.

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(c.) Reduce the following to decimal fractions, carrying out the division, when only approximate values can be obtained, to 6 places of decimals.

1ST SOLUTION.1⁄2 of a mile 7 m. 0 fur. 0 rd. 0 yd. 0 ft. 0 in.

to a decimal fraction.

or 154 of 1 = 14 of 5

14 of 500 hundredths

=

8..
9. 5.

10.3.

89. Fractional Parts of Denominate Numbers.

1. What is the value of of a mile in units of lower denomi

nations, i. e. in fur. rd. yd. etc.?

11. 26.

12. 12.

13. .

1 of 7 miles, which is found by dividing by 12, as in compound division.

12

of 8 fur.

=

2ND SOLUTION.-Since 1 m. = 8 fur., of a mile must equal 4 fur. But of a fur. of 40 rd. 26 rd. But of 5 yd. ofyd. 3 yd. But of a yd. Hence, of a mile 4 fur. 26 rd. 3 yd. 2 ft.

=

=

=

2.

of £1.

3.

of 1 gal.

4.

of 1 wk.

5. of 1 cord.

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3D SOLUTION. There must be 8 times as many furlongs as miles, or in of a mile, 8 times fur. 4 fur. But there must be 40 times as many rods as furlongs, or in fur. 40 times & rods 26 rods. But there must be 5 or times as many yards as rods, or in & of a rod

times yd.

3 yd. But there must be three times as many feet as yards, or in of a yard, 3 times & ft. 2 ft. Hence, of a mile 4 fur. 26 rd. 3 yd..

=

2

2 ft.

==

(a.) Reduce each of the following to units of a lower denomination:

2

of a rd.

13. .076 of a week.
14. .00276 of a league.

15. .1824 of a rood.

of 3 ft.

43

54

=

6.

7. of 1 ton.

IT

8. 1 of 1 mile.

17

9. of 1 degree.

8 of 1 acre. II

SOLUTION. There must be 12 times as many ounces as pounds, or, in this case, 12 times .5135 of an ounce = 6.1620 oz. 6.162 oz. But there must be 20 times as many pennyweights as ounces, or, in .162 of an ounce, 20 times .162 of a pennyweight = 3.240 dwt. = 3.24 dwt. But there must be 24 times as many grains as pennyweights, or, in .24 of a pennyweight, 24 times .24 of a grain 5.76 gr. Hence, .5135 of a lb. 6 oz. 3 dwt. 5.76 gr.

=

10. Reduce .5135 of a pound Troy to units of a lower denomination.

2 ft.

.5135

12 6.1620 20

(b.) Reduce each of the following to lower denominations:

11. .4595 of a £.

16. .00568 of a cord.

12. .3572 of a ton.

17. .475 of a bushel.

18. .396 of a lb.

NOTE.It will be seen that, practically, the reductions in the last example, as well as in those immediately preceding it, were made by multiplying the fractional part of each denomination by the number which expresses how many units of the next lower denomination are equal to a unit of the denomination of the fraction.

19. .0005 of a mile.

20. .1876 of an acre.

3.240 24 5.76

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