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10. Mr. Smith left $6,000 to his wife, $4,600 to his son and $4,000 to each of his two daughters, but upon settling his estate it amounted to only $12,400: how much did each receive ?

11. A and B engaged in a cotton speculation; A invested $4,800, and B $5,600 ; they lost f of their stock by fire, and gained on the remainder 4 of its cost : what was the gain of each ?

12. A, B and C engaged in business ; A furnished $4,000, B $6,000, and C managed the business; they gained $1,800 : what was the share of each, if received as much as A and B together ?

13. A merchant fails in buginess with debts amounting to $35,000, and resources to $16,500. He owes A $9,000, B $7,500, and C $11,000 : what should each receive?

14. Illustrate by an original problem the method of finding the gain or loss when the shares of the several partners are invested for unequal periods of time.

15. Illustrate by an original problem the method of finding the shares in the gain or loss of three partners in business, when their shares of capital are unequal and a full account of resources and liabilities is taken.

16. A and B formed a partnership; A put in $650 and B put in $1,050. After 6 months they take in C, who furnishes $1,000 capital. At the close of the year their profits are $2,000 : what is each man's share ?

17. A, B, and C formed a partnership for 3 years. They were each to have $100 a year as a salary. A put in $2,000 at the beginning, B $3,000, and C $2,500. After 2 years A drew out $500, B put in $500 more, and C put in $500 more. At the close of the 3 years their profits were $10,000: what was each man's share?

Arithmetical Analysis.

ART. 425.-An understanding of the conditions of a problem will make clear the method of solution.

In the analysis of problems all solutions are referred from one to many parts, and from many to one part. One whole or one part is the key to all solutions.

ORAL EXERCISES.

1. A can do a piece of work in 6 days and B in 9 days : in what time can both do the work together ?

Analysis.—Since A can do the work in 6 days, he can do t of it in one day ; and since B can do it in 9 days, he can do š in one day, and, in one day, both can do 5 + ý = p of the work. To do all the work will take them as many days as is is contained times in 1x = 33 days.

2. A can do a piece of work in 10 days and B in 20 days: in what time can both do the work together ?

3. A, B and C can do a piece of work in 6 days : A can do it alone in 18 days, and B can do it alone in 24 days : in what time can C do it alone ?

Analysis.-Since A, B and C can do t of the work in 1 day, and A and B can do the sum of ts and A in 1 day, C can do as much in 1 day as the difference between t and its + r'a) = . Hence C would require as many days to do the work as is contained times in 7%, which is 143 times, or 143 days.

4. A, B, and C can do a piece of work in 12 days, and A and B can do it in 20 days : how long will it take C to do ic ?

5. The head of a fish is 10 inches long. Its tail is as long as the head and half the body, and its body is as

long as the head and tail together : what is the length of the fish ?

Analysis.—The length of the head is 10 inches, the length of the tail is 10 inches + the body, and the body is 10 inches + 10 inches + f of itself. Then 20 inches must be the other { of the body and } of the body is 40 inches. The tail is 20 inches + 10 inches 30 inches, and the length of the fish is 10 + 30 + 40 inches 80 inches.

6. The head of a fish is 8 inches long; the tail is as long as the head plus } the body, and the body is as long as both the head and tail : what is the length of the body?

7. What is the time of day, if the time from now till noon is one-half of the time from noon to midnight ?

Analysis.—The time from noon to midnight is 12 hours, and I of that is 6 hours, which is the time from now till noon, or 6 A.M.

8. What is the time, if one-third of the time past noon is one-sixth of the time to midnight ?

Analysis.—The time between noon and midnight is 12 hours, which, if divided into two parts so that they are to each other as $ is to ļ, will give the time it lacks of being midnight, = 4 P.M.

9. Ten years ago I was 3 times as old as my son, and now I am 2 times as old : what are our respective ages ?

Analysis.—Ten years ago my son was one times his age, and now he he is 1 times his age at that time + 10 years, and I am 3 times his age at that time – 10 years. Twice his age + 10 years or 2 times + 20 years = 3 times his age + 10 years. Then 1 times his age 10 years ago

= 10 years.

10. A man dying left an estate worth $42,000. He provided that if his son returned from a foreign land he should receive twice as much as his wife, but if the daughter returned she should receive one half as much

as his wife : if both son and daughter returned, how should the estate be divided ?

Analysis.—By the conditions of the problem, the daughter should receive one half as much as the wife, and the wife one half as much as the son. Then the daughter should receive 1 part, the wife 2 parts, and the son 4 parts, or }, }, and 4, or $6,000, $12,000, and $24,000 respectively.

11. A hare is 100 leaps ahead of a hound and takes 4 leaps to 3 of the hound; but 2 of the hound's leaps equal 3 of the hare's leaps : how many leaps must the hound take to catch the hare ?

Analysis.-Since 2 of the hound's leaps equal 3 leaps of the hare, 3 leaps of the hound equal 4+ leaps of the hare ; but these 3 leaps of the hound are taken in the same time that the hare takes 4 leaps ; hence the hound gains į a leap in every 3 leaps, and he must take 6 leaps to gain 1 whole leap of the hare ; to gain 100 leaps, he will have to take 600 leaps.

12. A, B and C have $7,800 invested in business together, their interest being in the ratio of }, } and į. B dies, and the remaining partners pay his widow $3,900 cash for his interest in the business : what is each one's share of the remaining capital?

Analysis.-After paying for B's interest, the remaining capital is $7,800 $3,900 $3,900, which divided in the ratio of } to 1 gives $1,200 for A, and $2,700 for C.

13. A merchant bought goods at 36 cents a yard, which he sold at a net profit of 33% after allowing his customers a deduction of 20%: what were the goods marked ?

Analysis.—Since he gained 33% or }, he received 48 cents per yard, and since this is a discount of 20% or of his asking price, it is of his asking price, -- 60 cents.

14. When gold is worth 50% premium, how much gold is a dollar bill worth ?

Analysis.—Since a gold dollar is worth 150 cents in paper money, a dollar in paper money is worth 1:8, or }, = 663 cents in gold.

WRITTEN EXERCISES.

1. I bought 121 cords of wood at $41 per cord, and paid for it with flour at $53 per barrel : how many barrels were required ?

2. A man owning f of a factory, sold of his share for $4,000 : at that rate what was the value of the factory?

3. What must I ask per acre for land which cost $90, so as to sell it at 25% less than the asking price and still make 10%?

4. I sold goods at $2.40 per yard and lost 20%: for what price should I have sold it to gain 20% ?

5. A, B, C and D invested $25,000 in a mining speculation ; A's gain was $1,600, B's $2,400, C's $2,800, and D's $3,200 : what was the capital of each ?

6. E invested 50% of his money in a house and 25% of the remainder in furniture; the difference between the sums was $600 : what was his total investment ?

7. If 8 men plough 12 acres in 9 days, how many men can plough 8 acres in 12 days?

8. If 29 horses eat 186 bushels of oats worth $92 in 13 days, how long will it take 3 horses to eat one-half as much?

9. If 18 men can do a piece of work in 44 days, in how many days can they do twice as much work with the assistance of 8 more men ?

10. If 11 men can cut 147 cords of wood in y days by working 14 hours a day, in how many days can 5 men cut 150 cords of the same kind of wood, by working 10 hours each day?

11. A stock company has 360 shares equally divided among 9 stockholders. If A sells 10 shares to B and B sells of his stock to C, what % of A's share is B’s ?

12. A merchant bought corn at 30 cents a bushel. It

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