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A ratio may be either direct or inverse, simple or compound.

A Direct Ratio expresses the division of an antecedent by a consequent, and is the ratio usually meant when no direction is given to the contrary. Thus, if we speak of the ratio of 8 to 12, the direct ratio, or , is the one intended.

An Inverse Ratio expresses the division of a consequent by an antecedent. Thus, the inverse ratio of 8:12 is 12

A Simple Ratio consists of one antecedent and one consequent; as, 2 : 4.

A Compound Ratio consists of the product of two or more simple ratios. Thus, the simple ratios of 8 : 12 and 2 : 4 give the compound ratio 8 x 2 : 12 x 4.

In comparing fractional quantities, they must first be changed to a common denominator; when this is done, the fractions will be to each other in the ratio of their numerators. Thus, : =j&: = 30 : 32.

Since a ratio is expressed by a fraction, it follows that the terms of a ratio may be multiplied or divided in the same manner as the terms of a fraction, without changing the value of the ratio. Thus, the ratio 20 : 32 or may be changed to its lowest terms ķ, giving an equal ratio 5:8; or, it may be changed to any number of higher forms by multiplication, without altering the relative value of its terms.

ORAL EXERCISES. 1. What is the ratio of 8 to 4? ANALYSIS.—8:4=:=2; or, the ratio of 8 : 4 is 2.

2. 2 is what part of 10; or, what is the ratio of 2 to 10? What is the ratio of 10 to 2?

3. 4 is what part of 1; or, what is the ratio of 1 to 4? ANALYSIS. = 4, and 1 : 4 as 1 : 2=1. 4. is what part of 1? is what part of į? 5. What is the ratio of sto ?

6. If the antecedent is 4 and the ratio 2, what is the consequent?

ANALYSIS.-As the antecedent or dividend is 4, and the ratio or quotient is 2, the consequent or divisor must be 4; 2=2, and the terms are 4 : 2.

7. If the consequent is 8 and the ratio 2, what is the antecedent?

ANALYSIS.—If the consequent or divisor is 8, and the ratio or quotient is 2, the antecedent or dividend must be 8 X 2=16, and the terms of the ratio are 16 : 8.

8. The numerator of a fraction is 8, and the value of the fraction is 1; what is the denominator?

9. The denominator of a fraction is 16, and the value of the fraction is 1; what is the numerator?

10. Find the value of each of the following ratios:18:3, 3:18, 18: is, ģ: 4.

WRITTEN EXERCISES. 1. Change the ratios 3 : 7 and 4:5 to a simple ratio.

$X$= = 12 : 35. 2. If the consequent is and the ratio , what is the antecedent?

3. 1 a peck is what part of a bushel, or what is the ratio of 4 a peck to a bushel ?

4. What is the inverse ratio of 4:4? Note. As a direct ratio is found by dividing the antecedent by the consequent, an inverse ratio is found by dividing the consequent by the antecedent.

5. 25 cents are what part of $1.50, or, what is the ratio of 25 cents to $1.50?

6. f of a penny is what part of a shilling? ANALYSIS.—In one shilling there are 12 pence, the ratio therefore is : 12=1:46 or 3 : 48=*f or 16.

7. What is the ratio of 25 yards to 3 rods?
8. What is the ratio of 3 inches to 1 yard?

9. What part of £1 5 s. are 6 s., or what is the inverse ratio of £1 5 s. to 6 s.?

10. What part of 17 is 31 ? What is the ratio of 17:31? 11. What is the ratio of 1 : 167? 12. What part of 1 square foot are 14 square yards?

13. What is the ratio of $1 to £1, the custom-house value of £1 being $4.8665?

14. What part of $1 is £1?

PROPORTION.
A Proportion expresses an equality of ratios.

Proportion is indicated by placing four dots between the two ratios, or by placing the sign of equality between them; thus, 4:8:: 12:24, or 4:8=12: 24. This proportion may be read, 4 is to 8 as 12 is to 24, or, the ratio of 4 to 8 is equal to the ratio of 12 to 24.

The first and last terms of a proportion are called the Extremes, and the second and third terms the Means.

In any proportion the product of the means is equal to the product of the extremes. Thus, in the proportion 4:8 :: 12 : 24, 8 x 12 = 4 x 24.

A Simple Proportion expresses the equality of two simple ratios, consisting of four terms, any three of which being given, the fourth can always be found.

ORAL EXERCISES. 1. If 2 pounds of butter cost 50 cents, what will 6 pounds cost?

ANALYSIS.—The ratio of the pounds is 6 : 2, and the consequent of the second ratio is 50 cents; the proportion will therefore be 6: 2 :: what : 50. As the product of the means equals the product of the extremes, 6 x 50, or 300, is equal to 2 multiplied by the anteccdent 150, which is obtained by dividing 300 by 2; therefore 6 : 2 :: 150 cents : 50 cents. Cost of 6 pounds, $1.50.

2. 2:4 :: 6 : what? ANALYSIS.—The product of the means is 24, and the quotient of 24 = 2 is 12; therefore 2:4:: 6:12; or = 1.

3. 5:9 :: 10 : what? 4:8 :: 10 : what?

4. 2 : what :: 4:12? ANALYSIS.—The product of the extremes is 24, and the quotient of 24:- 4= 6; therefore 2: 6 :: 4:12; or = 4.

5. If 3 pounds of sugar cost 30 cents, what will 10 pounds cost? 3:10 :: 30 : what?

6. What : 6 :: 8 : 24? = how many 6ths?
9. If 4 yards of cloth cost $16, what will 5 yards cost?
8. 8:10 :: what : 5? = how many 5ths ?

9. The product of the means is 25, the first term is 5; what is the last term?

10. The product of the extremes is 24, the third term is 12; what is the second term?

11. If 3 barrels of apples cost $15, what will 12 barrels cost?

12. What : 6 :: 7: 21? 1=how many 6ths? 13. = how many 9ths? 9:27 :: what : 9?

14. The numerator of a fraction is 12, and the value of the fraction is 1; what is the denominator?

WRITTEN EXERCISES. 1. If 5 barrels of apples cost $20, what will 100 barrels

cost?

180 x 20 = $400. ANALYSIS.-As 100 barrels will cost more than 5 barrels, we express the ratio of 100 : 5, as an improper fraction 180, and multiplying by the remaining or third term, 20, we obtain 400 as the number of dollars that 100 barrels of apples will cost.

2. If $100 earn $6 in a year, what will $1500 earn in a year?

156000 x 6 = $90. ANALYSIS.-In this example it will be noticed that all three terms are of the same kind; but two of them are quantities which are said to earn something, whilst the third term is what is earned by one of the other two; we therefore make the two that earn, the terms of the ratio 1500 : 100, or 15.000, and multiply by the other term, 6, obtaining $90 as the amount that $1500 will earn in a year.

Note. When the first and second terms are of different denominations, they must be changed to the same denomination; and the third term must in like manner be changed to the lowest denomination named in it. The answer will always be of the same denomination as the third term.

RULE. Express the ratio of the terms which are of the same kind, as a proper fraction, when the answer requires to be less than the remaining term, and as an improper fraction, when the answer requires to be greater; multiply by the term which is of the same kind as the required answer, and the product will be the fourth term or answer.

3. If 10 gallons of wine cost $42.50, what will 63 gallons cost?

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