3. A and B can perform a piece of work in 6 days, A and C in 8 days, and B and C in 12 days. In how many days can each of them alone perform it ?

SOLUTION.

x = no. days in which A alone can perform it, y = no. days in which B alone can perform it,

z = no. days in which C alone can perform it. the part which A can perform in one day,

1

= the part which B can perform in one day,

the part which C can perform in one day. = the part A and B can perform in one day,

the part A and C can perform in one day,

the part B and C can perform in one day.

3300

(1) (2)

(3)

(4)

(5)

and

-1

-1

+y

1

1

x

+:

Adding (1), (2), and (3), 2x-1+2y-1+2z-1

|| || || | ||

=

=

x-1

y-1

(6)

(7)

(8)

Subtracting (1) from (5),

From (6),

z = 4'8

x= 93, no. of days in which A can perform it. 16, no. of days in which B can perform it. z = 48, no. of days in which C can perform it.

From (7), y =
From (8),

4. A says to B, if of my age were added to of yours, the sum would be 19 years. But, says B, if of mine were subtracted from of yours, the remainder would be 18 years. Required their ages.

Ans. A's age, 30 years; B's, 20 years.

5. If 1 be added to the numerator of a certain fraction, its value is; but if 1 be added to its denominator, its value is. What is the fraction? Ans. 15.

6. A farmer had 89 oxen and cows; but, having sold 4 oxen and 20 cows, found he then had 7 more oxen than cows. Required the number of each at first.

Ans. 40 oxen; 49 cows.

7. A says to B, if 7 times my property were added to of yours, the sum would be $990. B replied, if 7 times my property were added to of yours, the sum would be $ 510. Required the property of each.

Ans. A's, $140; B's, $70.

8. If of A's age were subtracted from B's age, and 5 years added to the remainder, the sum would be 6 years; and if 4 years were added to of B's age, it would be equal to of A's age. Required their ages. Ans. A's, 98 years; B's, 15 years.

9. It is required to divide 50 into two such parts that. of the larger shall be equal to of the smaller.

Ans. 32 and 18.

10. A gentleman, at the time of his marriage, found that his wife's age was to his as 3 to 4; but, after they had been married 12 years, her age was to his as 5 to 6. Required their ages at the time of their marriage.

Ans. The man's age, 24; his wife's, 18 years.

11. A farmer hired a laborer for ten days, and agreed to pay him $12 for every day he labored, and he was to forfeit $8 for every day he was absent. He received at the end of his time $40. How many days did he labor, and how many days was he absent?

Ans. He labored 6 days; was absent, 4.

12. A gentleman bought a horse and chaise for $208, and of the cost of the chaise was equal to the price of the horse. What was the price of each?

Ans. Chaise, $112; horse, $96.

13. A and B engaged in trade, A with $240, and B with $96. A lost twice as much as B; and, upon settling their accounts, it appeared that A had three times as much remaining as B. How much did each lose?

Ans. A lost $96; B lost $48.

14. Two men, A and B, agreed to dig a well in 10 days, but, having labored together 4 days, B agreed to finish the job, which he did in 16 days. How long would it have required A to dig the whole well? Ans. 16 days.

15. A merchant has two kinds of grain, one at 60 cents per bushel, and the other at 90 cents per bushel, of which he wishes to make a mixture of 40 bushels that may be worth 80 cents per bushel. How many bushels of each must he use?

Ans. 13 bushels at 60 cents; 26 at 90 cents.

16. A farmer has a large box, filled with wheat and rye; seven times the bushels of wheat are 2 bushels more than four times the bushels of rye; and the quantity of wheat is to the quantity of rye as 3 to 5. Required the bushels of wheat and the bushels of rye.

Ans. Wheat, 9 bushels; rye, 15 bushels.

17. My income and assessed taxes together amount to $50. But if the income tax be increased 50 per cent., and assessed tax diminished 25 per cent., the taxes will together amount to $52.50. Required the amount of each Ans. Income tax, $20; assessed tax, $ 30.

tax.

18. A and B entered into partnership, and gained $200. Now, 6 times A's accumulated stock (capital and profit) was equal to 5 times B's original stock; and 6 times B's profit exceeded A's original stock by $200. Required the original stock of each.

Ans. A's stock, $500; B's stock, $700.

19. A boy at a fair spent his money for oranges. If he had got five more for his money, they would have averaged a half-cent each less; and if three less, a halfcent each more. How many cents did he spend, and how many oranges did he get?

Ans. 30 cents; 15 oranges.

20. A merchant has three kinds of sugar. He can sell 3 lbs. of the first quality, 4 lbs. of the second quality, and 2 lbs. of the third quality, for 60 cents; or, he can sell 4 lbs. of the first quality, 1 lb. of the second quality, and 5 Ibs. of the third quality, for 59 cents; or, he can sell 1 lb. of the first quality, 10 lbs. of the second quality, and 3 lbs. of the third quality, for 90 cents. Required the price of each quality.

Ans. First quality, 8 cents per lb.; second, 7 cents; third, 4 cents.

21. A gentleman's two horses, with their harness, cost him $120. The value of the poorer horse, with the har

ness, was double that of the better horse; and the value of the better horse, with the harness, was triple that of the poorer horse. What was the value of each?

Ans. Harness, $50; better horse, $40; poorer, $ 30.

22. Find three numbers, so that the first with half the other two, the second with one third of the other two, and the third with one fourth of the other two, shall each be equal to 34. Ans. 10, 22, and 26.

23. Find a number of three places, of which the digits have equal differences in their order; and, if the number be divided by half the sum of the digits, the quotient will be 41; and, if 396 be added to the number, the digits will be inverted. Ans. 246.

24. There are 4 men, A, B, C, and D, the value of whose estates is $14,000; twice A's, three times B's, half of C's, and one fifth of D's, is $16,000; A's, twice B's, twice C's, and two fifths of D's, is $18,000; and half of A's, with one third of B's, one fourth of C's, and one fifth of D's, is $4000. Required the property of each.

Ans. A's, $2000; B's, $3000; C's, $4000; D's, $5000.

25. A and B driving their turkeys to market, A says to B, give me 5 of your turkeys, and I shall have as many as you. B replies, but give me 15 of yours, and then yours will be of mine. What number of turkeys had Ans. A, 45 turkeys; B, 55 turkeys.

each?

26. A person possesses a capital of $30,000, on which he gains a certain rate of interest; but he owes $20,000, for which he pays interest at another rate. The interest which he receives is greater than that which he pays by $800. A second person has $35,000, on which he gains the second rate of interest; but he owes $24,000, for which he pays the first rate of interest. The sum which he receives is greater than that which he pays by $310. What are the two rates of interest?

Ans. First rate, 6 per cent.; second rate, 5 per cent.