The use of $500 for 8 months is equivalent to the use of $4000 for 1 month; and the use of $600 for 10 months is equivalent to the use of $6000 for 1 month. Consider A's capital to be $4000 and B's $6000 A's share of gain, ; B's share of gain, 1.

6

2. A commenced business with $10000 capital. Four months later B put in $10500. Their profits at the end of a year were $5100. What was each man's share of the gain?

3. Three persons loaned a sum of money for which they received $1596 interest. The first had $4000 invested for 12 mo., the second $3000 for 15 mo., and the third $5000 for 8 mo. How much interest did each receive?

4. A and B were in partnership for 2 years. A at first invested $2000, and B $2800. At the end of 9 months A took out $700, and B put in $500. They lost in the two years $3720. Apportion the loss.

5. A, C, and H form partnership. A puts in $8000, C $5000, H $10000. A's capital remains in the business 8 mo., C's 9 mo., H's 12 mo. The net gain is $6900. Find each man's share of the gain.

6. Two partners entered business, agreeing to continue for 18 months. A put in $2000 at first, and 8 months later $1200 additional. B at first put in $3000, but at the end of 4 months drew out $600. On closing their account they found they had made $2808. What was each man's

share of the gain?

7. On Feb. 1 Messrs. Scott and White commenced business with $3000, each furnishing $1500. On April 1 White put in $1300 more. On May 1 they took Watson into partnership with $2500. At the close of the year, how should a net gain of $2400 be apportioned?

8. A's capital was in business 6 months, B's 7 months, and C's 11 months. A's gain was $600, B's $1400, and C's $990. Their joint capital was $7800. What was each man's capital?

9. A put $600 in trade for 5 months, and B $700 for 6 months. They gained $228. What was each man's share? 10. April 1, 1895, A goes into business with a capital of $6000; July 1, he takes in B as a partner with a capital of $8000; and Oct. 1, 1896, they have gained $2900. Find the gain of each.

11. Three men, A, B, and C, hire a pasture for 6 months for $75. A puts in 10 cows at first, but at the end of 1 month takes away 4. B puts in 8, and in 3 months takes out 5, but adds 2 after 2 months more. C puts in 6, and in 4 months he puts in 8 more. What should each pay?

QUESTIONS.

339.

1. Define ratio; the terms of a ratio.

2. How is ratio found? What is direct ratio?

verse ratio? A simple ratio?

A compound ratio?

In

3. Tell how to find ratio when antecedent and consequent are given. To find consequent when antecedent and ratio are given. To find antecedent when consequent and ratio are given. What are the principles of ratio?

4. Define proportion.

proportion? Name them.

How many terms in a simple

5. Give the principles of proportion.

6. What number is placed for the third term?

The

second? The first? How is the fourth term then found?

7. What is a compound proportion? What number is How is each couplet then

placed as the third term ?

arranged? How find the fourth term?

8. What is a partnership? A company?

9. Define capital stock; dividends.

10. Tell how to find each partner's share when the capital of each is invested for the same time. When the capital of each is not invested for the same time.

INVOLUTION.

340. 1. 3 x 4 x 2 = what?

2. 3 x 3 x 3 what?

NOTE.

=

In Ex. 2 the factors are equal; in Ex. 1 they are unequal.

The product of equal factors is a Power.

3. What is the product of 4 taken 3 times as a factor? 4. What is the product of 6 taken twice as a factor? 5. What is the product of used three times as a factor?

6. What is the product of .6 used twice as a factor? 341. The process of finding powers is Involution.

342. A power is named according to the number of its equal factors.

The product of two equal factors is the Second Power, or Square, of the equal factor.

The product of three equal factors is the Third Power, or Cube, of the factor.

NOTE. The second power is called a square because the area of any square figure is the product of two equal factors, length and breadth; and the third power is called a cube because the solidity of any cube is the product of three equal factors, length, breadth, and thickness.

343. A small figure at the right and above a number to show how many times it is to be used as a factor is called an Exponent. Thus,

42

23

3+

=

4 x 4 is 4 to the second power, or the square of 4;

2 x 2 x 2 is 2 to the third power, or the cube of 2;

3 × 3 × 3 × 3 is 3 to the fourth power, or the fourth power of 3.

Read: 82, 15, 57, (3/4)2, 3/42, 32/4, 8410, 168.

345. 1. What factor is used 3 times to produce 27?

2. What are the two equal factors of 64?

3. What is one of the three equal factors of 8? 4. 36 is the square of what number?

5. 64 is the cube of what number?

6. 144 is the second power of what?

7. 1728 is the cube of what?

346. One of the equal factors of a power is a Root.

One of two equal factors of a number is the Square Root of it.

One of the three equal factors of a number is the Cube Root of it.

The fourth root of a number is one of its 4 equal factors. The square root of 16 4. The cube root of 27

=

= 3.

347. The Radical Sign (√) placed before a number indicates that its root is to be found.

The radical sign alone before a number indicates the square root; thus, √9 3 is read, the square root of 9

=

3.

348. A small figure placed in the opening of the radical sign is called the Index of the root, and shows what root is to be taken; thus, V8 2 is read, the cube root of 8 is 2. Read the following:

=

√81, 64, 81, √144, 1728, 9, 3.64.

4. Write all the squares from 1 to 100.

5. Write all the cubes from 1 to 1000.

6. Learn the second and third powers of numbers from 1 to 12.

SQUARE ROOT.

350. The square of a number is the product of that number taken twice as a factor.

From the above illustration it is seen that annexing one cipher to a number annexes two ciphers to the square of

that number, as in 12:

=

1; 102

=

100; 1002 = 10000.

351. A square contains twice as many figures as its root, or twice as many less one.