Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

(e.) When the numerator of the unit of the compound fraction is not a multiple of the denominator of the other fraction, the preceding solution will give a fraction in the numerator of the result. In all such cases the following soluIndeed, applying, as it does, the principles of cancellation, it really includes the preceding. 15. Reduce of to a simple fraction.

tion should be adopted.

[ocr errors]

First Solution. Since of any fraction must equal 3 times as many parts, each as large as before, & of may be found by multiplying the denominator of by 8, and the numerator by 3. This gives the following written work.

[ocr errors][merged small]

Second Solution.

the denominator, and expressed by making

3

[blocks in formation]
[blocks in formation]
[blocks in formation]
[ocr errors]

6

[blocks in formation]

of

of

written work as before.

may be expressed by making 8 a factor of must be 3 times this result, which may be a factor of the numerator. This gives the same

NOTE. In writing the work of such examples as the above, the fraction on which the operation is to be performed should always be written first.

16. Reduce of 1 of 3 to a simple fraction.

8

First Solution. 31, or, being the number on which the operation is to be performed, should be written first. of this must equal 15 times as many parts, each as large, and may be expressed by making 15 a factor of the numerator, and 28 a factor of the denominator. of this must equal 8 times as many parts, each as large, and hence may be expressed by making 8 a factor of the numerator, and 9 a factor of the denominator. This would give the following work.

[blocks in formation]

Second Solution. 31, or, is the number on which the operation is to be performed, and should therefore be written first. of may be expressed by making 28 a factor of the denominator, and of

must be 15 times this result, which may be expressed by making 15 a factor of the numerator. of this result may be expressed by making 9 a factor of the denominator, and §, or 8 times the last result, by making 8 a factor of the numerator. This would give the same written work as before.

NOTE.- It will be seen that both of the above solutions give the same numerical process, viz., to 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.

Reduce each of the following fractions to simple ones.

[blocks in formation]

142. Multiplication of Fractions.

1. What is the product of

X?

Solution. multiplied by = of, which, found by multiplying the numerator of by 3 and the denominator by 8, gives the following written work.

[merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

(a.) By reading the sign X as times, we have

=

times of, which, found by multiplying the numerator of by 4 and the denominator by 9, gives the following written work.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][merged small]

(b.) The following form of explanation may be adopted when a very thorough analysis is required.

11. What is the product of

98

Solution. First write as the number on which the operation is to be performed; we then have multiplied by 1 equals, and multiplied by 's will equal g's of this result, which may be expressed by making 98 a factor of the denominator. If multiplying by gives this result, multiplying by must give 81 times this result, which may be expressed by making 81 a factor of the numerator. Hence,

[blocks in formation]

NOTE. It will be seen that all the forms of solution give similar forms of written work, the numerators of the several fractions being in all cases factors of the numerator of the product, and the denominators factors of the denominator.

[blocks in formation]

* For models of written work, see 171st page, solution to 8th example.

[blocks in formation]
[blocks in formation]

143. Reduction of a Vulgar Fraction to a Decimal Form Reduce to a decimal fraction.

Solution., or of 1, = of 7 of 70 tenths 8 tenths, with a remainder of 6 tenths. But 6 tenths 60 hundredths, and of 60 hundredths = 7 hundredths, with a remainder of 4 hundredths. But 4 hundredths 40 thousandths, and of 40 thousandths = 5 thousandths. Hence, } = .8 + .07 + .005 = .875

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

Solution. 2, or of 1,= 28 of 25 = z's of 250 tenths = 8 tenths, with a remainder of 26 tenths. But 26 tenths = 260 hundredths, and of 260 hundredths

28) 25.00

2.60

[ocr errors]

; &c. Hence the following written work.

.892857 Ans., or dropping the 28,

=

or of a millionth, we shall have the approximate value of 2.892857

.080

.0240

.00160

.000200

.000004

Proof. .892857 1000000 = 28.

[ocr errors]

NOTE.

[ocr errors]

= 892857 times 1000000 = 6259000 of

Compare the above solutions with 87, (c.)

(b.) The fractions may also be reduced by the following solutions:

, or units, 10 times as many tenths =

[blocks in formation]
[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[blocks in formation]
[blocks in formation]
[ocr errors]

25 X 10
28

tenths,

25 X 10 X 10

=

hun

28

25 × 10 × 10 × 10 × 10 x 10 x 10

28
14

7

millionths =

6250000
7

millionths.8928574

25

28

[blocks in formation]

25 × 10 × 10 x 10 x 10 x 10 x 10
28 × 10 x 10 x 10 x 10 x 10 x 10

14 ៗ

6250000 6250000

=

of .000001.892857+

[blocks in formation]

(c.) It is obvious that

equivalent decimal form, for introducing into the numerator

cannot be reduced to an exactly

« ΠροηγούμενηΣυνέχεια »