ILLUSTRATION.-The ratio 6: 7-7=1; or the ratio 6: 7 is the same as the ratio 1 14.

5. What is the ratio of 3 to 12? Of 8 to 13? Of 9 to 17 ?

6. What is the ratio of 5 to 6? Of 3 to 7?

7. What is the ratio of to 3?

ILLUSTRATION. The ratio of 2 to 3 is

is the same as the ratio: 43.

Of 7 to 3?

31

40

94; or the ratio 2 : 33

LESSON II.

PROPORTION.

276. A Proportion is an equality of ratios; thus, the ratio 9: 3 is equal to the ratio 12: 4; hence 9: 3 and 12: 4 form a proportion; thus, 9: 3=12: 4.

277. A Proportion is usually expressed by four dots; thus, 3:7 :: 6 : 14, to be read 3 is to 7 as 6 is to 14.

278. Terms. Every proportion consists of four terms; the first and fourth are called the extremes; the second and third are called the means; the first and third are called the antecedents; the second and fourth are called the consequents.

(a.) Principles.-I. If the antecedents or consequents of a proportion, or both, are multiplied or divided by the same number, the proportion will still

exist.

ILLUSTRATION.-Dividing the antecedents of the proportion 4: 8 :: 10:20 by 2, we have 2: 8 :: 5:20. Dividing the consequents, we have 4: 4 :: 10: 10. Dividing both, we have 2: 4 :: 5:10. Multiplying the consequents, we have 4:16 10:40. Multiplying the antecedents, we have 8: 8:: 20:20. Multiplying both, we have 8: 16 :: 20:40. Dividing the first couplet, we have 2:4: 10:20. Dividing the last couplet, we have 4:8:: 5:10. Each of these is a proportion, for the ratios of each are equal.

II.—In every proportion, the product of the extremes is equal to the prod uct of the means.

III.-Either mean is equal to the product of the extremes divided by the other mean.

IV. Either extreme is equal to the product of the means divided by the other extreme.

If 9 pounds of coffee cost $3.60, what will 11 lb. cost?

MODEL OPERATION.

Extreme. Mean. Mean. Extreme.

9 lb. 11 lb. :: $3.60: cost 9 lb. : 11 lb. :: $3.60 : $4.40

11

9)39.60

$4.40, the required extreme.

FORM.-1. Since, by the conditions of the question, 9 lb. cost $3.60, 11 lb. will cost as much as $3.60, hence the proportion 9:11 :: $3.60: cost of 11 lb.

2. Since in every proportion the product of the extremes is equal to the product of the means, the required extreme may be found by dividing the product of the means 11 and $3.60, by the given extreme 9 (278. IV.) which gives $4.40 for the required extreme.

Therefore, if 9 lb. of coffee cost $3.60, 11 lb. will cost $4.40.

(a.) Rule.-From the conditions of the question arrange the terms so that the ratios will be equal; then (278) find the required term.

1. If 11 bushels of wheat cost $9, what will 17 bus. cost? 2. What will 7 gallons of molasses cost, if 63 gallons cost $13.16?

3. A fox is 35 rods before a greyhound, and while the fox is running 2 rods the greyhound runs 25 rods; how far must the dog run, before he can catch the fox?

COMPOUND PROPORTION.

279. Compound Proportion is an equality be.tween a compound and a simple ratio, or between two compound ratios.

(a.) A compound ratio is the product of two or more simple ratios.

If 8 men can cut 40 acres of wheat in 3 days, how many acres can 9 men cut in 4 days?

MODEL OPERATION.

8:9 :: 40: (A.)
3:45

and 24:36 :: 40: (a.)

or 2:3 : 40: (60 s.)

3

120÷2=60.

FORM.-1. Since by the conditions of the question, 8 men can cut 40 acres, 9 men can cut as much as 40 acres; hence the proportion 8:9::40: to the required number of acres.

2. Since in 3 days 40 acres of wheat can be cut, in 4 days 3 as much as 40 acres can be cut, hence the proportion 3: 4 :: 40: to the required number. 3. Since the ratio of the first couplet of the proportion is 8: 9, and the ratio of the first couplet of the next proportion is 3:4; the compound ratio (279. a.) is 24:36, and the proportion becomes 24: 36 :: 40 : (4.), or (278. a.) 2:3:: 40 (4.); the required term of which is found as in simple proportion. (278. IV.)

(a.) Rule.-I. From the conditions of the question find by analysis the ratio of each of the couplets; then the product of the simple ratios will give the compound ratio.

II. Find the required term as in simple proportion.

4. If 12 men can mow 11 acres of grass in 5 days, how many men can mow 33 acres in 18 days?

5. 480 lb. of flour will last 24 men 40 days, how long will 300 lb. last 48 men, at the same rate?

6. 144 men in 6 days of 12 hours each, can dig a ditch 200 ft. long, 3 ft. deep, and 2 ft. wide; in how many days. of 7 hours each, can 30 men dig a ditch 350 ft. long, 6 ft. deep and 3 ft. wide?

LESSON III.

Any example that can be solved by Proportion may be easily solved by analysis.

If 20 men in 164 days of 8 hours each, dig a ditch 88 rods. long, 8 feet deep, and 3 feet wide; how many men will be required to dig a ditch 360 rods long, 12 feet deep, and 8 ft. wide, in 18 days, by working 12 hours each day?

Men.

MODEL OPERATION.

Men.

20 × 33 × 8 × 1 × 1 × 1 × 360×2×××1200

Anal. Steps. (a.) To dig a ditch in 16 days, according to the condi tions of the question, 20 men are required.

1. One day will require of 33 times as many men as 16 days.

2. One hour will require 8 times as many men as 8 hours.

3. One rod long will require

4. One foot deep will require

as many men as 88 rods.

as many men as 8 feet.

5. One foot wide will require as many men as 3 feet.

6. 360 rods long will require 360 times as many men as one rod.

7. 12 feet deep will require 12 times as many men as one foot.

8. 8 feet wide requires 8 times as many men as one foot.

9. 18 days will require as many as one day.

10. 12 hours will require as many men as one hour, which is 200 men.

7. A man bought 753 bu. of potatoes at 95 ct. a bu. and paid for them in cloth at $1.37 ct. per yd.; how many yd. did it take?

8. 4 men are paid $32 for 6 days labor; how many men may be employed 21 days, for $84 ?

9. 7 hhd. of molasses cost $250.75; how much will § of a hogshead cost?

10. 16 men can gather 44 acres of grain in 12 days by working 10 hours a day; how many men will be required to gather 440 acres in 10 days, by working 16 hours a day?

11. 12 men can cut 36 cords of wood in 8 days by working 10 hours a day; how many cords can 50 men cut in 26 days by working 12 hours a day?

12. In a certain factory 9448 yards of cloth were made by employing 120 hands 78 days, 10 hours a day; how much more could be made by employing 300 hands 4 times as many, days for 10 hours 10 minutes each day?

13. I have a marble slab which is 25 feet long, 5 feet wide, and 31⁄2 inches thick, and weighs 825 pounds; what will be the weight of a slab whose length is 8 feet, width 3 feet, and thickness 3 feet?.

14. 8 men are paid $48 for 6 days' labor; how many men may be employed 16 days for $96 ?

15. If 6 men can dig a ditch 80 feet long, 6 feet wide, and 4 feet deep, in 15 days, in what time can 18 men dig one 240 feet long, 8 feet wide, and 6 feet deep?

16. If 8 men pay $136.20 for 5 weeks' board, how much must 8 men pay for 3 weeks' board?

17. The walls of a fortification are to be raised to the

height of 27. feet; a squad of 12 men can raise them 9 feet in 6 days: how many men will be required to finish them in 4 days?

18. 5 bushels of wheat are worth 12 bushels of rye; 8 bushels of rye are worth 20 bushels of oats; and 9 bushels of oats are worth $4: how many bushels of wheat will $50 buy? 19. If 8 yards of cloth 14 yds. wide cost $50, what will be the cost of 14 yds. of cloth of the same quality, 24 yds. wide?

20. I have a piece of coal which is 2 feet long, one foot wide, and 6 inches thick, and it weighs 55 pounds; what will be the weight of a block 110 feet long, 92 feet wide, and 24 feet thick?

21. I have a stone 4 inches wide, 8 inches long, 3 inches thick, weighing 2 lbs.; what will be the weight of a monument, of the same kind of stone, 30 feet long, 12 feet wide, and 12 feet thick?

22. If $500 will gain $30.12 in 4 mo. 12 days, at 9 per cent., how much will $750 gain in 2 yrs. 9 mo. 8 days, at 6 per cent.?

SECTION XIV.

INVOLUTION AND EVOLUTION.

LESSON I.

INVOLUTION.

280. Involution is the method of finding any power of a given number.

281. A Power of any number is a quantity produced by taking that number a certain number of times as a factor.. The factor thus taken is called the root of the power.

282. The Exponent or Index is the number denoting the power, and is a small figure placed above the root at the right; thus, 9' indicates the third power of 9, or 9 × 9 × 9; (3)* indicates the fourth power of 3.

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