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Fractions.

187. To subtract one mixed number from another when the fraction in the subtrahend is greater than the fraction in the minuend.

EXAMPLE.

From 583 take 32.
Operation.

Explanation.
58% = 5821 19 is greater than a, therefore we take 1 unit
32j = 328

from the 8 units, change it to 24ths, and add it

to the 9 24ths. Difference 2521

ift=. - 19 = it. 2 units from 7, (8 — 1), units = 5 units. 3 tens from 5 tens = 2 tens.

I. Find the difference of-1. 24% and 163

6. 354 and 261 2. 29 and 151

7. 283 and 143 3. 4638 and 1877 8. 362and 8180 4. 525 and 313

9. 654 and 223 5. 47% and 18%

10. 343 and 271 (a) Find the sum of the ten differences.

II. Reduce to simplest form

1. 5; + 3} – 51
2. 6% - 3 + 41
3. 2 - 11 + 3%
4. 73 + 3- 13
5. 6; 31 + 5}
6. 31 + 43 - 33 - 2 + 13
7. 6% - 2 - 13 - 111 + 21

8. 52 + 43 + 2% + 3} + 31
(b) Find the sum of the eight results.

Fractions.

188. To multiply a fraction by an integer.

Multiply ak by 6.
Operation No. 1.

Operation No. 2.
6 times a are = 1

6 times a d = 1 = 1; 1. Observe that by the first operation we obtain ji ; that in 13 there are 6 times as many parts as there are in and that the parts are of the same size as those in 1:

2. Observe that by the second operation we obtain ; that in there are the same number of parts as there are in 3, and that the parts are 6 times as great as those in 74.

Note 1.—The 7 of 14 may be regarded as a dividend; the 24, as a divisor, and 4 itself, as a quotient. In 14, we have a dividend 6 times as great as that in za, the divisor remaining unchanged. In 1, we have a divisor 1 sixth as great as that in 1, the dividend remaining unchanged. Multiplying the dividend or dividing the divisor by any number, multiplies the quotient by the same number.

Note 2.—The 7 of 3 may be regarded as the antecedent of a couplet ; the 24, as the consequent, and in itself as the ratio. Multiplying the antecedent or dividing the consequent of any couplet multiplies the ratio by the same number.

RULE. - To multiply a fraction by an integer, multiply its numerator or divide its denominator by the integer. I. Find the product.

1. x4 5. X8 9. 12 x 4
2. i x 6 6. } x 9 10. I x 6
3. 11 x 5 7. x 8 11. 70 5
4. 37 x 7 8. x9 12. 13 x 7
(a) Find the sum of the twelve products.

II. Find the product.
1. 37 x 7 5.4% x 5

9. 6.3 x 5
2. 5% 6 6. 17 x 4 10. 2:3 x 4
3. 7,6 x 4 7. 53 x 5 11. 45 x 6
4. 3.7
5

8. 85 x 4 12. 61 x 7
(b) Find the sum of the twelve products.

Х

Fractions.

189. To divide a fraction by an integer.

Divide & by 3.
Operation No. 1.

Operation No. 2.
One third of q =

One third of 1
One third of 5 = 1

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1. Observe that by the first operation we obtain ; that in there are 1 third as many parts as there are in and that the parts are of the same size as those in .

2. Observe that by the second operation we obtain ; that in there are the sanie number of parts as there are in and that the parts are 1 third as great as those in .

Note 1.-The 6 of g may be regarded as a dividend; the 7 as a divisor, and the itself as a quotient. Inį, we have a dividend i third as great as that in ,, the divisor remaining unchanged. Ini, we have a divisor 3 times as great as that in %, the dividend remaining unchanged. Dividing the dividend or multiplying the divisor by any number, divides the quotient by the same number.

RULE. - To divide a fraction by an integer, divide its numerator or multiply its denominator by the integer.

1.

7

I. Find the quotient. (See p. 105, problems 15 and 16.)
+4
4. 4

7. 11 + 4
2.
5. & + 5

8. 17 + 5
3. i20

6. ő = 20

9. 21 (a) Find the sum of the nine quotients.

7 12

• 20

II. Find the quotient. (See p. 105, problems 17 and 18.) 1. 17% + 3

4. 18 % + 3 7. 164 = 3 2. 17 + 4

5. 18 30 + 4 8. 16+ + 4 3. 17} + 6 6. 1836 + 6 9. 164 + 6

(b) Find the sum of the nine quotients.

Fractions.

190. To MULTIPLY BY A FRACTION. $6 multiplied by 3, means, take 3 times $6. $6 x 3 = $18.

. $6 multiplied by 2, means, take 2 times $6. $6 x 2 = $12. . $6 multiplied by 21, means, take 21 times $6; or 2 times $6

+ of $6. $6 x 24 = $15. $6 multiplied by 1, means, take of $6. $6 x 4 = $3. $6 multiplied by f, means, take of $6. $6 x š = $4.

TO TAE TEACHER.-Require the pupil to examine the preceding statements and similar ones presented by the teacher or by himself, until he clearly understands that to multiply by a fraction is to take such part of the multiplicand as is indicated by the fraction. Thus: to multiply 48 by { is to take three fourths of 48; that is, three times I fourth of 48. It will thus be clear that multiplication by a fraction involves both multiplication and division; hence the work on the preceding pages should be mastered by the pupil before attempting what follows.

EXAMPLE I.
Multiply 24 by .
1 fourth of 24 is 6.
3 fourths of 24 are 18.

EXAMPLE II.
Multiply by s.
1 fourth of is do
3 fourths of are no.

EXAMPLE III.

EXAMPLE IV. Multiply 275} by .

Multiply 346% by 24.* 1 fourth of 275% is 68 %.

Two times 316% = 692. 3 fourths of 275% are 206%. 1 half of 346% = 1733.

6924 + 173} = 866 Ans. RULE. - To multiply by a fraction, divide the multiplicand by the denominator of the fraction and multiply the quotient thus obtained by the numerator of the fraction.

Observe that in practice we may, if more convenient, multiply the multiplicand by the numerator of the fraction, and divide the product thus obtained by the denoniinator. To multiply 12 by we may take 3 times 1 fourth of 12 or 1 fourth of 3 times 12, as we choose

*This means, take 2 times 3463 and of 3463.

Fractions,

1. Find the product. (See p. 105, prob. 19 and 20.) 1. 345 x 3

4. 263 x 1 7. 263 x 1
2. 345 x ** 5. 263 x

8. 576 x
3. 345 x }
6. 263 x

9. 576 x
(a) Find the sum of the nine products.

3

II. Find the product. (See p. 105, prob. 21 and 22.)
1. io x Š
4. š x }

7. x1
2. to x ot
5. x

8. } x ģ
3. * }
6. x

9. <3
(b) Find the sum of the nine products.

[II. Find the product. (See p. 106, prob. 23 and 24.)

1. 3721 x 1 4. 5233 x } 7. 523 x 1
2. 3721 x po

5. 5233 x 8. 153x
3. 3724 x 6. 523 x

9. 153, (c) Find the sum of the nine products.

IV. Find the product. (See p. 106, prob. 25 and 26.) 1. 4623 x 210

6. 3463 x 3 2. 4623 x 316

7. 3463 x 21 3. 4623 x 2

8. 2754 x 41 4. 346] x 2

9. 2754 x 31 5. 346} x 33

10. 2754 x 2 (d) Find the sum of the ten products. (For a continuation of this work see page 101.) * Take 3 times 1 tenth of 345, or 1 tenth of 3 times 345.

+ Lead the pupil to see that in problems of this kind, the correct result may be obtained by multiplying the numerators together for a new numerator and the denominators together for a new denominator”; that in so doing he divides the multiplicand by the denominator of the multiplier and multiplies the quotient so obtained by the numerator of the multiplier,

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