2. Compared with $8 taken as the standard, what is the relative value of $16? Of $40? Of $56? Of $72? Of $96? Of $1? Of $3? Of $4? Of $6 ?

[a] The consequent of any direct ratio is the standard or BASIS of comparison [divisor].

[b] The antecedent of any direct ratio is the quantity compared with the standard [dividend].

[c] The relative value of the antecedent [quantity compared] as measured by the consequent [standard of comparison] is the DIRECT ratio [quotient].

[d] To reverse a and b, that is, to take the standard as the antecedent [dividend] and the quantity compared as the consequent [divisor], will produce an INVERSE ratio [inversed quotient].

17. What is the inverse ratio of $23.4375 to $3.125 ?

18. What is the direct ratio of 23.1 bu. to 4.2 bu.?

19. The consequent is 715.35 miles and the antecedent 5722.8 miles. What is the ratio?

20. How much greater or less is the ratio of 17.125 cords: 102.75 cords than the ratio of 17.75 cu. ft.: 115 cu. ft.?

21. What is the difference between the inverse ratio of 7.92 miles: 47.52 miles and the direct ratio of 3 feet: 30 feet?

22. The antecedent is 5 years and the consequent 2 years. What is the ratio?

When the ratio and one of its terms are given, to find the other term.

35. If the ratio is 51, and the antecedent 52 days, what is the consequent?

36. If the reciprocal of the ratio of two numbers is 7, and the consequent oz., what is the antecedent?

37. The reciprocal of the consequent is 5 weeks, and the reciprocal of the ratio 54. What is the antecedent?

38. The inverse ratio of two quantities is 134, and the standard of comparison 7 gal. What is the quantity compared ?

To find the value of a compound ratio.

[j] Value of compound ratio = product of all the antecedents ÷ product of all the consequents.

To find an omitted term of a compound ratio.

[k] (Value of compound ratio X product of all the consequents) + product of given antecedents = omitted antecedent.

[1] (Product of all the antecedents + value of compound ratio) + product of given consequents = omitted consequent.

EXAMPLES.

Find the omitted term in the following compound ratios:

1. What term of division corresponds with the ratio? With the antecedent? With the consequent? What is the ratio of $18: $6? How was this ratio obtained? What quantity divided by $5 will produce a quotient equal to this ratio? Hence, what quantity has the same relation or ratio to $5 as $18 has to $6?

2. What quantity has the same ratio to 4 lb. as $6 has to $3? As 8 gal. has to 2 gal.? As 9 bu. has to 3 bu.? As 15 yr. has to 3 yr.? As 24 cd. has to 8 cd. ? 3. What is the ratio of $4 to $2? 10 lb. divided by what quantity will produce a quotient equal to this ratio? Hence, the antecedent $10 has the same relation to

what consequent as $4 has to $2 ?

4. The ratio of 12 lb. to 3 lb. is equal to the ratio of 8 gal. to what quantity? Of 24 cwt. to what quantity? Of 40 yr. to what time? Of 36 mi. to what distance ? Of 28 cd. to what quantity?

[a] A proportion denotes that the relative value [ratio] of one antecedent when compared with its consequent equals the relative value [ratio] of another antecedent when compared with its consequent.

[b] The product of the extremes of a proportion equals the product of its means. [c] The product of the extremes of a proportion divided by one of its means equals the other mean.

[d] The product of the means of a proportion divided by one of its extremes equals the other extreme.

EXAMPLES.

Find the omitted term represented by x.

1. 4 oz. 9 oz.:: x: 25 cts.

2. $3 x 9 yd.: 12 yd.

:

9.

{

30 men : 5 men

4 da. : 8 da.

{75 rd.: 25 rd.

bu.

3 mi.

9 bu.

x 1

28 hr.: 6 hr.

9 da.: 15 da.

}

:: $160: $90.

:: $28.80: $18.

$18: x.

::x:11 men.

SIMPLE PROPORTION.

INDUCTIVE EXERCISE.

1. How many times greater are 6 lb. than 2 lb. ? Hence, if 2 lb. of sugar cost 20 cents, what should be the cost of 3 times 2 lb., or 6 lb., of the same sugar? How does the ratio of 6 lb. to 2 lb. [the respective weights] compare with the ratio of 60 cents and 20 cents [their respective cost]?

2. If 3 yd. of cloth cost $6, what will be the cost of 6 yd. of the same cloth? Of 12 yd.? Of 18 yd.? Of 21 yd.? Of 27 yd.? Of 36 yd.?

[TO THE TEACHER.]—In each of the foregoing and following exercises the pupil should be required to compare the ratio of the quantities considered with the ratio of their cost.

3. What part of 24 lb. of flour are 8 lb. of flour? Hence, if 24 lb. of flour cost 96 cents, what should be the cost of of 24 lb., or 8 lb. ? How does the ratio of 24 lb. to 8 lb. [the respective weights] compare with the ratio of 96 cents to 32 cents [their respective cost]?

4. If 18 lb. of tea cost $9, what will be the cost of 9 lb. of the same tea? Of 6 lb? Of 3 lb? Of 2 lb?

[a] The second term of a proportion has the same relation to the first that the fourth has to the third.

[b] (Third term × second term) ÷ first term = fourth term.

EXAMPLES.

1. What will be the cost of 13 pounds of sugar if 8 pounds cost 72 cents?

2. If 12 pounds of beef cost $1.80, how much will 9 pounds cost? 3. A farmer raised 1750 bushels of wheat on 70 acres of land. At the same rate how many bushels could he have raised on 87 acres?

4. If 50 bushels of wheat cost $75, what will 80 bushels cost at the same rate?

5. If a field of 300 acres will produce 16500 bushels of corn, what will a field of 93 acres produce at the same rate?

6. If 24 men can do a given piece of work in 9 days, how long will it require 18 men to do it?

7. If 25 pounds of coffee are worth 5 pounds of tea, how many pounds of tea are worth 35 pounds of coffee? How many pounds of coffee are worth 3 pounds of tea?

8. If I am charged $12.25 for 35 yards of goods, how much should I be charged for 47 yards of the same goods?

9. How many oranges will cost as much as 180 lemons, if 20 lemons cost as much as 15 oranges?

10. If a garrison consume 102 barrels of flour in 17 days, how many barrels of flour will it consume in one year?

11. What will 30 pounds of butter cost, if 18 lb. 3 oz. cost $5.82? 12. If 82 sheep cost $287, how many sheep at the same price will cost $52.50?

13. If $320 will accrue $9.60 interest, how much interest will $240 accrue in the same time and at the same rate %?

14. In weighing out 176 pounds of sugar, a retail grocer lost 21 pounds by wastage. At that rate, how much would he have lost in weighing out 260 pounds?

15. If of a yard of goods cost of a dollar, what part of a dollar will of a yard cost?