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Fractions.

189. To divide a fraction by an integer.

Divide by 3.
Operation No. 1.

Operation No. 2.
One third of g 24

One third of 4 = d'1
One third of

6 21

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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 4.

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

Note 1.—The 6 of ļ 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. In it, we have a divisor 3 times as great as that in %, the dividend remaining unchanged. Dividing the dividend or multiplying the divisor by any pumber, 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.

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(a) Find the sum of the nine quotients.

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

4. 18 % + 3 7. 16} 3
2. 17
4

5. 18 % + 4 8. 165 + 4
3. 17 + 6
6. 18.36 + 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 2} = $15. $6 multiplied by }, means, take į of $. $6 x } $3. $6 multiplied by , means, take of $6. $6 x = $4.

TO THE 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 4.
1 fourth of 24 is 6.
3 fourths of 24 are 18.

EXAMPLE II.
Multiply by a.
1 fourth of is
3 fourths of fare zo.

EXAMPLE III.

EXAMPLE IV. Multiply 275 by .

Multiply 346% by 23.* 1 fourth of 275% is 686.

Two times 316% = 692%. 3 fourths of 275% are 206, 1 half of 346% = 173}.

692 + 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 denominator. 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 346; and of 346%.

Fractions,

I. Find the product. (See p. 105, prob. 19 and 20.)
1. 345 x 2
4. 263 x }

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

8. 576 x Š
3. 345 x }
6. 263 x

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

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

7. x1
2. mox pot 5. x Š

8. x
3. :? X}
6. x

9. La
(b) Find the sum of the nine products.

III. Find the product. (See p. 106, prob. 23 and 24.)
1. 3724 x 1
4. 523 x }

7. 523 x 1
2. 3724 x 10 5. 5233 x 8. 153] x
3. 3721 x 6. 523 x 9. 1534 x

(c) Find the sum of the nine products.

IV. Find the product. (See p. 106, prob. 25 and 26.) 1. 462 x 20

6. 346 x 3 2. 4623 x 36

7. 3464 x 27 3. 4623 x 23

8. 2754 x 41 4. 3464 x 2}

9. 2753 x 34 5. 3464 x 33

10. 2754 x 24 (d) Find the sum of the ten products. (For a continuation of this work see page 101.) * Take 3 tinies 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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Observe that in cases like the above, in which the denominators are prime to each other, the 1. c. d. is the product of the given denominators, and each new numerator may be found by multiplying the given numerator by the denominator of the other fraction.

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Assign a numerical value to each letter and verify.

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SOLVE.—Then let x = 5 and y = 7 and verify.

4.

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Algebraic Fractions.
192. Problems in Multiplication and Division.

Example 1.

2

6
x 3

1}
b
6

5 5

a

ac

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ac

6

Let a = 2, b = 5, c = 3, and d = 7; then

bd

2 x 3
5 x 7

35

I. Find the product and verify as above.
a?

3.
ху

5.
63

x 3 a bx

ах

1.

хс

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Observe that in the verification of these problems any value you please may be given to a letter only provided it retains the value given it, throughout the solution of that particular problem.

II. Find the quotient and verify as above.

a?
1.
63

3.
y

5.

• 3 a y?

bx

х

ах

.2

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