T

date, pay $1500 down, how long in equity should I be permitted to keep the remainder after the expiration of the 4 months? Ans. 6 months.

10. A hare starts 25 of its leaps in advance of a hound, and takes 4 leaps to the hound's 3; but 2 of the hound's leaps are equal to 3 of the hare's; how many leaps must the hound take to overtake the hare?

SOLUTION. If 2 of the hound's leaps equal 3 of the hare's, 1 of the hound's is equal to 11⁄2 of the hare's, and 'f the hound takes 3 leaps to the hare's 4, he takes 1 leap to the hare's 1.

If 1 of the hound's leaps is equal to 11⁄2 of the hare's, and he takes 1 leap to the hare's 1, he gains in taking 1 leap, 11⁄2 — 1}, or } of a hare's leap.

If he gains of a hare's leap in taking 1 leap, he will gain 25 leaps of the hare, or overtake the hare, in taking as many leaps, as is contained times in 25, which are 150.

Therefore, etc.

11. A thief having gone 51 miles, an officer set out to overtake him, and for 16 miles traveled by the thief, the officer travels 19 miles. How far will the officer have traveled before the thief is overtaken? Ans. 323 miles.

12. A starts from Boston toward a town 16 miles distant, walking at the rate of 24 miles an hour; and 2 hours after, B starts from Boston upon the same route, by coach driven at the rate of 9 miles an hour; in what time, and how far from Boston will B overtake A?

Ans. In 40 minutes, and 6 miles from Boston. 13. If 12 barrels of corn will pay for 10 cords of wood, and 48 cords of wood will pay for 8 tons of hay, how many barrels corn will pay for 15 tons of hay?

of

SOLUTION. If 8 tons of hay can be paid for by 48 cords of wood, 15 tons can be paid for by 15 of 48 cords, or 90 cords.

If 10 cords of wood can be paid for by 12 barrels of corn, 90 cords

REVIEW QUESTIONS. What is the Rule for finding the equated balance of an account? (346) What is Merchandise Balance? (347) Interest Balance? (348) Cash Balance? (349)

♦f wood, or 15 tons of hay, can be paid for by 9 times 12 barrels, or 108 barrels.

Therefore, etc.

14. If 10 calves are worth as much as 9 colts, and 90 colts are worth as much as 112 sheep, how many sheep are worth as much as 50 calves? Ans. 56.

15. If 8 men can do the work of 32 women, and 2 women can do the work of 3 boys, how many men can do the work of 24 boys? Ans. 4 men.

16. If the relative value of oak wood to spruce is as 2 to 1, and that of spruce to pine as 7 to 8, how many cords, composed of spruce and pine in equal parts, will equal 60 cords of oak? Ans. 112 cords.

17. A, B, and C together can do a piece of work in 20 days, A alone can do it in 60 days, and B alone can do it in 80 days; in what time could C working alone do it?

SOLUTION. If A, B, and C together can do the work in 20 days, in 1 day they can do of it.

If A working alone can do the work in 60 days, in 1 day he can do of it, and if B working alone can do it in 80 days, in 1 day he can do ; hence, A and B working together can do + 3%, or 240, of the work in 1 day.

of the work in 1 day, since of it in 1 day, C must do the of the work in 1 day.

If A and B working together can do A, B, and C working together can do difference between 2 and 40, or 240 If C can do of the work in 1 day, working alone, he can do the whole of it in as many days as is contained times in 1, or in 48 days. Therefore, etc.

18. A piece of work can be done in a day of 11 hours by 2 men, or 5 women, or 12 boys; in what time could it be done by 1 man, 2 women, and 3 boys, working together?

Ans. In 10 hours.

REVIEW QUESTIONS. What are Taxes? (353) A Poll Tax? (353) How is a town tax assessed or apportioned? (357) What are Duties? (358) Internal Revenue? (361) Customs? (363) Exchange? (369) Domestic Exchange? (376) Foreign Exchange? (379)

23

19. A carpenter is offered $325 to do a job of work, which he can do in 12 days, his journeyman in 183 days, and his apprentice in 25 days. If they should do it together, in what time could it be completed, and how much would each earn? Ans. In 5 days; the carpenter $150, the journeyman $100, and the apprentice $75.

INVOLUTION.

385. A Power of a number is either the number itself, or the product obtained by taking the number several times as a factor. Thus,

212, is the first power of 2.

22

=

2 X 2 = 4, is the second power, or square of 2. 282 X 2 X 2 =

8, is the third power, or cube of 2. 242 × 2 × 2 × 2 16, is the fourth power of 2,

=

and so on, the exponent (Art. 103) of the power denoting the number of times the number 2 is taken as a factor.

386. Involution is the process of raising a given number to a required power.

This may be effected, as is evident from the definition of a power, by the simple process of multiplying. That is, to raise a number to any power,

Find the product of the number taken as a factor as many times as is denoted by the exponent of the required power.

Exercises.

Raise to the powers denoted by their respective exponents:

What is a Power? What does the exponent of the power denote? What

is Involution?

How is a number raised to any power?

9. What is the square of 78? 10. What is the cube of .09?

Ans. 6084.

Ans. .000729.

387. From the definition of a power may be derived the following

PRINCIPLES.

1. The product of two or more powers of a number is that power denoted by the sum of their exponents.

For, since 23 × 22: = (2 × 2 × 2) × (2 × 2) = 25, 2o × 22 = 23+2

= 25.

2. Any power of a number raised to a power is that power of the number denoted by the product of the exponents. For, since (23)2 = (2 × 2 × 2) × (2 × 2 × 2) = 2o, (23)2 = 23×a =2o.

3. Any power of a number divided by a power of the same number is that power denoted by the difference of their expo

nents.

For, since 2522 = (2 × 2 × 2 × 2 × 2) ÷ (2 × 2) = 23, 25 ÷ 22 =25—2—2o.

Exercises.

1. What power of 2 is 25 X 22?

2. What is the product of 38 × 33?

3. What is the 7th power of 2?

Ans. 27. Ans. 729.

SOLUTION. 27 = : 24 × 23 = 16 × 8=128, Ans.

4. What is the 8th power of 5?

Ans. 390625.

5. What power of 15 is 152 raised to the 3d power?

6. Find the 2d power of 53.

7. What power of 17 is 175÷ 174?

8. Find the quotient of 106 ÷ 10% 9. What is the 8th power of ? 10. Required the value of (48). 11. Required the 10th power of 3.

What is the first Principle? The second?

Ans. 15°. Ans. 15625.

Ans. 171. Ans. 100. Ans. 6536 Ans. 262144.

Ans. 59049.

The third ?

EVOLUTION.

388. A Root of a number is one of the equal factors taken to form the number.

389. The Second, or Square Root of a number is one of its two equal factors. Thus,

The square root of 25 = 5; since 5 × 5 = 25.

The Third, or Cube Root of a number is one of its three equal factors.

Thus,

The cube root of 125 = 5; since 5 × 5 × 5 = 125.

390. The Radical Sign, ✅, or Fractional Exponents, are used to denote roots. Thus,

2

34, or 4, denotes the second or square root of 4 ;

3

4

8, or 8, denotes the third or cube root of 8;

or 16, denotes the fourth root of 16;

✓16, or

and so on, the figure or figures, called the Index, written over the radical sign, or the denominator of the fractional exponent, denoting the degree or name of the roots.

The index is usually omitted in denoting the square root. 391. Evolution is the process of finding or extracting the roots of numbers. It is the reverse of Involution.

A Perfect Power is a number whose root can be exactly obtained; and an Imperfect Power, or Surd, a number whose root cannot be exactly obtained.

392. The Root corresponding to any perfect power may be found by factoring the power.

Thus,

By resolving 196 into its prime factors, 2, 2, 7, and 7, we find its square root is 14, since one of its two equal factors is 7 X 2 = 14.

8

√2700030; since the factors of 27000 are 2, 2, 2, 3, 3 3, 5, 5, 5; and one of the three equal factors is 2 × 3 × 5 = 30.

What is a Root? The Square Root of a number? The Cube Root! By what are roots denoted? The degree or name of the roots? What is Evolution? A Perfect Power? An Imperfect Power?