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ing ratios: 18 to 4; 4 to 18; 15 to 9; 25 to 15; 105 to 45; 800 to 150.

3. What fraction expresses the ratio of 7 to 8? 24? Of 2 to 3? Of 4 to 6? Of to ? 43

NOTE. Reduce the pare the numerators.

of 15 to 18.

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27 to 45;

Of 15 to

fractions to a common denominator, and comThus, the ratio 5 to 18 is the same as that

4. Express in a common fraction the ratio of ratio of of to ; of to 7; off to 31; of 4 9 3

to 7.

to; the

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A proportion consists of two equal ratios. When four numbers are so related to each other, that the first has the same ratio to the second that the third has to the fourth, they constitute a proportion. Thus the numbers 4, 5, 12, 15, form a proportion, because the ratio of 4 to 5 is equal to the ratio of 12 to 15. The proportion may be expressed thus: 4:5= 12: 15; or, 4:5::12; 15; or = 1; which is read, 4 is to 5 as 12 is to 15; or, 4 divided by 5 is equal to 12 divided by

15.

The first and fourth terms of a proportion are called the extremes; and the second and third, the means. When the proportion is expressed in a fractional form, the numerator of the first fraction and the denominator of the second are the extremes, and the denominator of the first and the numerator of the second the means. In every proportion the product of the extremes is equal to the product of the means. In the above proportion 4:5=12: 15, or =1; the product of the extremes, 4 X 15, is equal to the product of the means, 5 X 12.

As the product of the extremes is always equal to the product of the means, we see that if the product of the means be divided by one of the extremes, (97,) the quotient will be the other extreme; and if the product of the extremes be divided by one of the means, the quotient will be the other

mean.

In questions in simple proportion, there are always three numbers or terms given, to find a fourth term or answer. Two

of the given terms are of the same name or kind, and the other given term is of the same name or kind as the answer.

EXAMPLE. If 3 barrels of flour cost $18, how many dollars will 8 barrels cost?

In this example the dollars should have the same ratio to each other that the barrels have to each other; viz., the ratio of 3 to 8. The proportion is written 3: 8=18: Ans. ; and is read, 3 barrels is to 8 barrels as $18 is to the answer. And, since the product of the means divided by one extreme gives 8 X 18 144 48; or, by cancelling,

the

other extreme,

6

8 × 18

=

3

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= 48, the 4th term, or answer.

Find the unknown terms in the following expressions. Cancel, if possible, before performing the work.

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2. 4.-; 31: 2071-; 151: 18}=25}

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;

: 1.3; 75: 1.3 = :

5. 5.19:5; 3.01: -=}:}; *: -=6.5. : 5.

6.5 18:25; : 7.5 15: 18; -18=

5:9.

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=

$15: $ ; 6 lb. 11 lb.

7. 3 barrels: 8 barrels: cts. : ; 4 lb.*: 5 oz. = 15 cts. :

:

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NOTE. When the terms of a ratio are of different denominations, they must be reduced to the same denomination.

3

(8.) 4 bu. 9 bu. 3 pk. $2.50; 3 yd. 2 qr.: 4 yd. qr.$12.50: $

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(9.) yd. qr.= $3:$—; lb. 33 lb.-$11⁄21⁄2: —;

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128. Since a proportion consists of two equal ratios, and as ratio is the relation of two quantities of the same kind only, the third term must always be of the same kind as the fourth term or answer; and the second must be either greater or less than the first, as the answer or fourth term is to be greater or less than the third term. Hence the following

RULE FOR SIMPLE PROPORTION.

Write the given number which is of the same kind as the required fourth term or answer, for the third term of the proportion. Then consider whether the answer is to be greater or less than the third term; if it is to be greater, write the greater of the two remaining terms for the second, and the other for the first term; but if it is to be less, write the less of the two remaining terms for the second term, and the other for the first.

Divide the product of the second and third terms by the first; the quotient will be the fourth term or answer.

The first and second terms must be of the same denomination; the fourth will be of the same denomination as the third term.

The learner should solve the following questions both by analysis (47) and by the rule for proportion.

1. If 9 yards of cloth cost $45, what will 15 yards cost?

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As the answer is to be in dollars, we make $45 the third term; and as 15 yards will cost more than 9 yards, the second term must be larger than the first.

To perform this question by analysis, say, if 9 yards cost $45, 1 yd. will cost, and 15 yards of $45=$75.

2. If 6 men do a piece of work in 20 days, how long will it take 15 men to do it?

NOTE. As 15 men will do it in fewer days than 6 men, the less of these two numbers must be the second term.

3. If 8 acres cost $98.50, what will 360 acres cost? 4. If 54 acres cost $2160, what will 9 acres cost?

5. If 7 men do a piece of work in 25 days, in what time will 5 men do it?

6. If 5 men do a piece of work in 35 days, in what time will 7 men do it?

7. How many men will it take to do in 25 days the work that 5 men will do in 35 days?

S. If 7 pairs of boots cost $24.50, how many pairs will $94.50 buy?

9. If 7 pairs of boots cost $24.50, how much will 27 pairs

cost?

10. If of a barrel of flour cost $2.70, what will 17 barrels cost?

11. If 17 barrels of flour cost $107.10, how much will $2.70 buy?

12. How many yards of cloth yd. wide will line a cloak containing 10 yards that are yd. wide?

1

13. If a cubic foot of water weighs 1000 oz., how many pounds of water will a cistern contain that is 3 ft. wide, 5 ft. long, and 5 ft. high? How many in a cistern 4 ft. wide, 3 ft. high, and 6 ft. long?

14. If the interest on a note at 6 per cent. is $125.15, what would be the interest at 5 per cent.?

15. If the interest at 6 per cent. were $52.95, at what rate per cent. would it be $61.771?

16. If the rate per year is 7 per cent., in what time would it be 24 per cent.?

17. If a post 6 feet high casts a shadow of 73 feet, on level ground, how high is a steeple which at the same time casts a shadow of 187 ft.? How long a shadow will a pole 60 ft. high cast?

18. Bought 48 yards of broadcloth for £33 12s. What are 27 yards worth at that rate? How many yards will £24 10s. buy?

19. If of a hhd. of molasses cost $12, what cost of a hhd. ?

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By Analysis.

If

cost

of $12.

cost $12, would cost, and would of a hhd. would cost of of $12= $16,

Ans.

20. If of a bushel is worth 24 cents, what is of a bushel worth?

21. If 3 lb. of pork cost 38 cts., what are 15 lb. worth? 22. If 8 men mow 333 acres of grass in 18 days, in how many days will they mow 284 acres?

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23. How much carpeting 1 yd. wide will cover a floor 5 yards long, and 3§ yards wide?

24. If 8 men can mow a field in 3 days, by working 10 hours per day, how long will it take them if they work only 9 hours per day?

129. COMPOUND RATIO.

A Compound Ratio is the ratio of the product of two simple ratios.

The ratio of

The ratio of

8 to 5 is
4 to 3 is

The ratio compounded of these is 32 to 15==&×$.

A compound ratio is reduced to the form of a simple ratio by multiplying the corresponding terms together.

1. Reduce to a simple form each of the following compound ratios.

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3:5

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130. COMPOUND PROPORTION.

Compound Proportion is the equality of two ratios, one of which is compound and the other simple. Thus, 3:8 is a compound proportion. After reducing the compound ratio to a simple one, the proportion becomes a simple proportion.

4:7

Reduce the following compound proportions to simple ones, and find the unknown term of the last ratio.

If the antecedents or first terms have factors common to the consequents or second terms, or to the third term, they should be cancelled before multiplying and dividing,

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