11. Three men engaged in business. A furnished $6000 and B $8000. They gained $4200, of which C's share was $1400. What was the gain of A and B and C's stock?

12. Five men trade in partnership. A furnishes $500, B $600, C $800, D $1000 and E $1200 capital. They gain $2750. What is the gain of each partner?

13. A, B and C bought a farm in partnership. A paid the purchase money, B and C the rest. They sold it at a gain of $3000. What was each one's share of the profit?

CASE II.

472. When the capital of the partners is employed for different periods of time.

WRITTEN EXERCISES.

1. A began business with $6000 capital. At the end of 6 months he took in B as a partner, who furnished $5000 additional capital. If the gain, after 6 months more, was $3400, what was each partner's share of the gain?

$30000 for 1 month. Both together had invested sums of money which were equal to the use of $102000 for 1 month, of which A contributed a sum equal to $72000 for 1 month, or 172, and he was therefore entitled to 2 of the gain, or $2400. B, contributed a sum

10

and was therefore entitled to

RULE.-Find such a part of the entire gain or loss, for each partner's share of the gain or loss, as the capital of each partner for a unit of time, is of the entire capital for a unit of time.

2. A engaged in business with a capital of $4000. After 3 months he took in B with a capital of $6000, and in 6 more, C became a partner, with a capital of $8000. At the end of 18 months the profits were $9360. What was each partner's share of the gain?

3. A, B and C engage in business together. A puts in $4000 capital for 8 months, B $6000 for 7 months, and D $3500 for one year. If they gain $2320, what is each partner's share of the gain?

4. B, C and D entered into partnership, furnishing a joint capital of $5875, of which B furnished 20%, C 35%, and D the rest. B's capital was employed 15 months, C's 9 months, and D's 10 months. They lost $2502.75. What was each partner's loss?

5. A, B and C took a contract to build a block of stores. A furnished 20 men for 3 months, B 25 men for 3 months, and C 15 men for 4 months. After paying the expenses the profits were $1475. What was the share of each? 6. A, B and C lost $8500 by speculating in real estate. A furnished $5000 of the capital which was employed for 1 year, B $8000 for 10 months, and C $10000 for 6 months. What was each one's share of the loss?

7. A, B and C engaged in manufacturing rope and cordage. A invested $4500 for 6 months, B $5000 for 8 months, and C $6500 for 7 months. They gained $4500. What was the gain of each partner?

8. G, L and F entered into partnership. G furnished. $1200, L $1500, and F $3000. After 6 months F withdrew $2000 of his capital. If at the end of a year the profits were $2200, what part of the profits belonged to each partner?

473. 1. A was employed on a piece of work 6 days, and B 12 days on the same work. How does the number of days A worked compare with the number of days B was employed?

2. A laborer earned $12 a week, and spent $6. How does what he spent compare with what he earned?

3. How does $3 compare with $9? $4 with $12? $6 with $18?

4. How does 2 compare with 10? 3 with 18? 5 with 25?

5. What relation has 2 to 12? 3 to 21? 4 to 28? 6. What is the relation of 3 to 24? 6 to 30? 7 to 35? 7. How does 8 compare with 2? What is the relation of 8 to 2?

8. How does 9 compare with 3? What relation has 9

to 3?

9. What relation has 24 to 8? 30 to 6? 10. What is the relation between 5 and 7? 11. What is the relation between 6 and and 9?

12. What is the relation of 8 to 9? 13. What is the relation of 12 to 4? 14. What is the relation of 15 to 5? 15. What is the relation of 16 to 8? 16. What is the relation of 25 to 5?

Between 8 and 9?
Between 12 and 4?
Between 15 and 5?
Between 16 and 8?
Between 25 and 5?

DEFINITIONS.

474. Ratio is the relation of one number to another of the same kind.

1. This relation is expressed either as quotient of one number divided by the other, and is called Geometrical Ratio, or simply Ratio, or, as the difference between two numbers, and is called Arithmetical Ratio.

2. When it is required to determine what the relation of one number to another is, it is evident that the first is the dividend, and the second the divisor.

3. When it is required to determine the relation between two numbers, either may be regarded as dividend or divisor.

4. The first number is commonly regarded as the dividend.

475. The Terms of a Ratio are the numbers compared.

476. The Antecedent is the first term.

Thus, in "What is the ratio of 6 to 8?" 6 is the antecedent.

477. The Consequent is the second term.
Thus, in "What is the ratio of 6 to 8?" 8 is the consequent.

478. The Sign of ratio is a colon (:).

Thus, the ratio of 12 to 6 is expressed, 12 : 6.

The colon (:) is sometimes regarded as the sign of division without the line. Thus, 12 : 8 is regarded as 12÷8.

479. The antecedent and consequent together form a Couplet.

480. PRINCIPLES.-1. The terms of a ratio must be like numbers.

2. The ratio is an abstract number.

3. Multiplying or dividing both terms of a ratio by the same number does not change the ratio of the numbers.

EXERCISES.

481. 1. What is the ratio of 3 to 6? 2. What is the ratio of $3 to $10?

bush. to 9 bush.?

5 to 10? 7 to 21? 12 lb. to 6 lb.? 27

3. What is the ratio of 7 to 35? 24 to 48? 13 to 39? 4. If the antecedent be 20, and the consequent 15, what is the ratio?

5. What is the ratio when the antecedent is 45, and the consequent 25?

6. What is the ratio of to ? to? to?

Fractions should be reduced to similar fractions. They will then have the ratio of their numerators.

7. What is the ratio of 51 to 31? 7 to 61? 9 to 51? 8. What ratio will the work of 12 men sustain to that of 8 men?

9. What will be the ratio of 8 yd. to 24 yd.? 63 yd. to 9 yd.?

10. When the antecedent is 3 and the ratio, what is the consequent?

11. When the consequent is 8 and the ratio, what is the antecedent?

12. When the antecedent is 1⁄2 and the ratio, what is the consequent?

13. What number has to 3 the ratio of 5 to 6?

14. What number has to 5 the ratio of 4 to 12?

15. What number has the ratio to 12 that 3 has to 1?

16. If two numbers have the relation of 6 to 8, and the first is 12, what is the other?

17. What number has to 12 the ratio of 8 to 9?

18. If two numbers have the relation of 10 to 15, and the antecedent is 40, what is the consequent?