Add 1 again and it becomes 1 + 4 r + 6 p2 + 4 μ3 + r1. This is now the 4th power of 1 + r, and it may be written (1 + r)*. Subtract the 1 which was added last, and it becomes (1 + r) — 1. Divide this by r, because it was multiplied by r, and it be comes (1 + r) * — 1 r Subtract 1 again, because 1 was added previous to multiplying by r; and it becomes (1+r)1 — 1 1 = (1 + r)' — (1 + r) — (1 +r) [(1 +r)3—1] = r Substitute t in place of the exponent 3, and multiply by a, and it becomes The particular question given above may now be solved by logarithms, using this formula. 2. A man deposited annually $50 in a bank from the time his son was born, until he was 20 years of age; and it was taken out, together with compound interest on each deposit at 3 per cent., when his son was 21 years of age, and given to him. How much did the son receive? 3. How much did the bankers gain by receiving the money, supposing they were able to employ it all the time at 6 per cent. compound interest? 4. A man has a son 7 years old, and he wishes to give him $2000 when he is 21 years old; how much must he deposit annually at 4 per cent. compound interest, to be able to do it? 5. If a man deposits in a bank annually $35, in how long a time will it amount to $500 at 6 per cent. compound interest? 6. The first slaves were brought into the American Colonies in the year 1685. Suppose the first number to have been 50, and that 50 had been brought each year for 100 years, and the rate of increase 3 per cent. How many would there have been in the country at the end of the hundred years? LIV. Annuities. 1. A man died leaving a legacy to a friend in the following manner; a sum of money was to be put at interest, such that, the person drawing 10 dollars a year, at the end of 15 years the principal and interest should both be exhausted. What sum must be put at interest at 6 per cent. to fulfil the above condition? Let the learner generalize this example and form a rule; and then solve the following examples by it. 2. A man wishes to purchase an annuity which shall afford him $300 a year so long as he shall live. It is considered probable that he will live 30 years. What sum must he deposit in the annuity office to produce this sum, supposing he can be allowed 3 per cent. interest? N. B. The principal and interest must be exhausted at the end of 30 years same, viz. 3 shillings, and the sum received by the first and second was 36 shillings less than that received by the third and fourth. How much did each receive? 12. There are two numbers, the greater of which is three times the less; and the sum of their second powers is five times the sum of the numbers. What are the numbers? 13. What two numbers are those, of which the less is to the greater as 2 to 3; and whose product is six times the sum of the numbers? 14. There are two boys, the difference of whose ages is to their sum as 2 to 3, and their sum is to their product as 3 to 5 What are their ages? 15. A detachment of soldiers from a regiment being ordered to march on a particular service, each company furnished 4 times as many men as there were companies in the regiment; but these being found insufficient, each company furnished three more inen, when their number was found to be increased in the proportion of 17 to 16. How many companies were there in the regiment? 16. Find two numbers which are in the proportion of 8 to 5, and whose product is 360. 17. A draper bought 2 pieces of cloth for $31.45, one being 50 and the other 65 cents per yard. He sold each at an advanced price of 12 cents per yard, and gained by the whole $6.36. What were the lengths of the pieces? 18. Two labourers, A and B, received $43.85 for their wages; A having been employed 15, and B 14 days; and A received for working four days $3.25 more than B for 3 days. What were their daily wages? 19. Having bought a certain quantity of brandy at 19 shillings per gallon, and a quantity of rum exceeding that of the brandy by 9 gallons, at 15 shillings per gallon, I find that I paid one shilling more for the brandy than for the rum. How many gallons were there of each? 20. Two persons, A and B, have each an annual income of $1200. A spends every year $120 more than B, and at the end of 4 years the amount of their savings is equal to one year's income of either. What does each spend annually? 21. In a naval engagement, the number of ships taken was 7 more, and the number burnt was 2 fewer, than the number sunk; 15 escaped, and the fleet consisted of 8 times the number sunk. Of how many did the fleet consist? 22. A cistern is filled in 50 minutes by 3 pipes, one of which conveys 10 gallons more, and the other 8 gallons less than the third, per minute. The cistern holds 1820 gallons. How much flows through each pipe per minute? 23. A farm of 750 acres is divided between three persons, A, B, and C. C has as much as A and B both, wanting 10 acres; and the shares of A and B are to each other in the proportion of 7 to 3. How many acres has each? 24. A certain sum of money being put at interest for 8 months, amounts to $772.50. The same sum put out at the same rate for 15 months amounts to 792.1875. Required the sum and the rate per cent. 25. From two casks of equal size are drawn quantities which are in the proportion of 5 to 8; and it appears that if 20 gallons less had been drawn from the one which now contains the less, only as much would have been drawn from it as from the other. How many gallons were drawn from each? 26. There are two pieces of land, which are in the form of rectangular parallelograms. The longer sides of the two are in the proportion of 6 to 11, and the adjacent sides of the less are in the proportion of 3 to 2. The whole distance round the less is 135 yards greater than the longer side of the larger piece. Required the sides of the less, and the longer side of the greater. 27. A person distributes forty shillings amongst fifty people, giving some 9d. and the rest 15d. each. How many were there of each ? 28. Divide the number 49 into two such parts, that the quotient of the greater divided by the less, may be to the quotient of the less divided by the greater as to 2. 29. A person put a certain sum to interest for 5 years, at 6 per cent. simple interest, and found that if he had put out the same sum for 8 years at 4 per cent. he would have received $60 more. What was the sum put out? 30. A regiment of militia containing 830 men is to be raised from three towns, A, B, and C. The contingents of A and B are in the proportion of 3 to 5; and of B and C in the proportion of 6 to 7. Required the numbers raised by each. 31. At what time between 6 and 7 o'clock are the hour and minute hands of a watch together? 32. There is a number consisting of two digits, the second of which is greater than the first; and if the number be divided by the sum of the digits, the quotient will be 4; but if the digits be inverted and that number divided by a number greater by 2 than the difference of the digits, the quotient will be 14. Required the number. 33. There is a fraction whose numerator being tripled, and the denominator diminished by 3, the value becomes ; but if the denominator be doubled and the numerator increased by 2, its value becomes. Required the fraction. 34. A merchant bought a hogshead of wine for $100. A few gallons having leaked out, he sold the remainder for the original sum, thus gaining a sum per cent. on the cost of it, equal to twice the number of gallons which leaked out. How many gallons did he lose? 35. There are two pieces of cloth, differing in length 4 yards; the first is worth as many shillings per yard as the second contains yards; the second is worth as many shillings per yard as the first contains yards; and both pieces are worth £72. 10s. How many yards does each contain? 36. A merchant bought a piece of cloth for $180, and selling it at an advance of $1 a yard on the cost, he gained 15 per cent. Required the number of yards. 37. There are two rectangular pieces of land, whose lengths are to each other as 3:2, and surfaces as 5:3; the smaller one is 20 rods wide. What is the width of the other? 38. There is a cistern to be filled with a pump, by a man and a boy working at it alternately; the man would do it in 15 hours, the boy in 20. They filled it in 16 hours 48 minutes. How long did each work? 39. In a bag of money there is a certain number of eagles, as many quarter eagles, the number of half eagles, together |