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7. Find the sum of of 16, and of 2 of 27. 16, and of 3 of 278, then 6 and

81

Thus of

reduced to a

common denominator, will become 128 and 243 which added together, make 3-15

Ans.

Ans. 20.

Ans. 33. Ans. 31.

8. Add together of 25, and of 152. 9. Find the sum of 5 and 2 of 5 of 13. 10. Add together,, and 7 of 3 of 51. Fractions of different integers (as found in money, weight, &c.) must be reduced to those of the same, before adding; or the value of each fraction may be found by Problem VIII. page 101, and then added together.

11. Add of a shilling to of a pound. Thus, s. reduced to the fraction of a pound, by Prob. IV. page 171, equal to £; then, £, and £ added together=40 of a pound, whose value by Prob. VIII. page 94,-4s. 9d. 3 qrs, Ans. Or thus, s. Os. 4d. 2qrs.

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RULE.

Prepare the fractions as in addition; then the difference of the numerators written over the common denominator, will be the difference of the fractions required,

EXAMPLES.

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1. Frod take of. Thus, of, then and reducised to their least common denominator by Rule 2,

page 169, become 21 and 12, and the difference between the numerators, 21 and 12, is 9, which written over the common denominator, becomes 24=3 Ans.

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take 5.

Ans. 191

450

7. From Note. To subtract a fraction from a whole number, take the numerator from its denominator, and place the remainder over the denominator, then take 1 from the whole number.

To subtract mixed numbers without reducing them to improper fractions. When the lower fraction is greater than the upper one, subtract its numerator from the common denominator, and to the difference add the upper numerator, then carry 1 to the lower whole number.

8. From 16 subtract 5. Thus, taken from leaves, that is, the numerator 5, subtracted from the denominator 9, leaves 4, which placed over the denominator is, then 1 taken from 16, leaves 15. Ans. 154. Ans. 24.

9. From 25 take 13.

10. From 29 take 153; thus, and reduced to a common denominator.

8

make and 12 then 29-2912 15=1512

Ans. 131

3

The lower fraction being greater than the upper one, we proceed thus; subtracted from leaves 2 and (the upper fraction) added to 11 (that is, we subtract the numerator of the

3

12

lower fraction (9) from its denominator (12) and to the difference add the upper numerator (8) makesh 11=11) we then carry 1 to the whole number, and subtract, and the rem. is 1311.

11. What is the difference between 145 and 193.

12. From 36 take 918.

Ans. 478 Ans. 26

8

MULTIPLICATION OF VULGAR FRACTIONS.

RULE.

Reduce Compound fractions to simple ones, and whole and mixed numbers to improper fractions. Then multiply the numerators together for a new numerator, and the denominators together for a new denominator

EXAMPLES.

1. What is the product of 5 multiplied by of? thus, 5-6, and of, then X-23. Ans. 2. Multiply by.

3. Multiply

by 3.

4. Multiply 4 by 1.

5. Multiply of by

of.

6. Multiply of 5 by of 8.

7. Multiply 7 by 91.

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8. Multiply 123 by of 7.

51

Ans. 12=1.
Ans. 15
Ans.

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DIVISION OF VULGAR FRACTIONS.

RULE.

Prepare the fractions as before, then invert the divisor and proceed exactly as in multiplication; and the products will be the quotient required.

How many times is contained in § ?

Thus, by the rule we invert the divisor 3, which becomes ; then x2=15, or 213, the answer.

45

5

5

Illustration of the Rule.-Had it been required to divide by 2 a whole number, instead of 2 ninths, it is evident it would have given the quotient 21 eighths, or ; but the divisor, being ninths, will be contained in the dividend 9 times were the whole number is contained 1 time; therefore the quotient is 9 times too small, and must be multiplied by 9, the denominator of the dividing fraction, and 9 16-18, or 218 the answer as before. This process consists only in multiplying the numerator of the divisor into the denominator of the dividend, and the denominator of the divisor into the numerator of the dividend.

5

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2. Divide 17 by 3

EXAMPLES.

Thus, &, Ans.
Quotient-12.
Quotient

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Ans. 188=138.

3. Divide of by 3 of 3.

4. What is the quotient of 79 divided by § of 7?

5. Divide 9 by 15.

6. Divide 45 by of 4.

7. Divide of 4 by 45.

8. How many times is 35 contained in

35

75

Quotient 961
Quotient 2.

Quotient
of 27?
Ans. 12195
Ans. 317
Quotient 19

9. How many times is contained in 48? 10. Divide 19 by 37. 11. At of a dollar per bushel, how many bushels of oats can be bought for $195, or $12?

Ans. 441bu.

12. At of a dollar per bushel, how much corn can be bought for 273 dollars?

Ans. 31 bushels.

RULE OF THREE DIRECT IN VULGAR FRACTIONS,

RULE.

Prepare the fractions as in multiplication: then state the question in the same manner as taught in the Rule of Three in whole numbers; then invert* the first term and multiply all three of the terms continually together, and the product will be the answer in the same name of the second or middle term.

EXAMPLES.

1. If of a barrel of flour cost of a dollar, what will of a barrel cost?

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5

bbl.

Thus, 3. Then the divisor or first term being inverted will stand thus xx3-$120, and $120 = 5dollars, the answer.

The same, by analysis.-If cost of a dollar 1 barrel,

*The reason of inverting the first term in this rule is very evident, since fractions depend on the same principle as whole numbers, and this produces the same effect as that of multiplying the second and third terms together and dividing by the first; therefore, the first term being a divisor, we invert it as taught in Division of Vulgar Fractions

RULE OF THREE DIRECT IN VULGAR FRACTIONS. 177

will cost 8 times as much. Thus, 8 times =40; then of $40-$120, or 5 dollars, the answer as before.

2. If of a bushel of wheat cost of a dollar, what will 8 bushels cost?

Thus,::, and xx11-861=$108, Ans. 3. If 10 dollars will buy 8 bushels of wheat, how much wheat will of a dollar buy?

=

80

bu.

41

bu.

Thus, 1061867, and 83-4; then 61::::, Ans. 4. If of a yard cost of a pound, what will of an Ell English come to?

syd. of of =20, or Ell English.

Ell. £ Ell.

4

15

£

105

s. d. qrs.

Then, as::; and 2×4×2×72=13 82. 5. If of a cwt. cost of a dollar, what will 15 cwt. come to? Ans. $20, 30c. 8m.+ 6. If of of an acre of land cost $9, what will 301

acres come to ?

Ans. $2041,871

32

7. If of a vessel cost 1236 dollars, what are 3 of her worth? Ans. $206. 8. A merchant sold 5 pieces of cloth, each containing 123 yards, at 9s. d. per yard; what did the whole come Ans. £31 9s. 10d. 31qrs. 9. At $3 per cwt., what will 92lbs. come to?

to?

2688'

Ans. $84, or 31cts. 28m.+ 10. A person owning of a vessel sold of his share for 835 dollars; what was the whole vessel worth at that rate ? Ans. 1565 dollars 62 cents.

RULE OF THREE INVERSE IN VULGAR FRACTIONS.

RULE.

Prepare the given fractions, and state the question as in direct proportion; then invert the third term and multiply all three of the terms together, the product will be the answer in the same name of the middle term.

EXAMPLES.

1. How much flannel that is of a yard wide, will line 5 yards of cloth which is 14 yards wide?

yds. wide. yds.

yards.

As 15; and 3×23×3=345-143 Answer.

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