and it may Add 1 again and it becomes 1 +4r + 6 m2 + 4 mx + pl. This is now the 4th power of 1 +r, 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 1 Subtract 1 again, because I was added previous to multiplying by r; and it becomes (1+r)-1 -1=(1+r)^(1+r) = (1+r)[(1+r):-1] Substitute t in place of the exponent 3, and multiply by a, and it becomes a (1+r) [(1 + r) — 1) = A.. r r r which is the same as before. The particular question given above may now be solved by logarithms, using this formula. log. (1+r) = 1.05 0.021189 Multiply by t = 15 15 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 3. If the man mentioned in the last example should die at the end of 18 years, how much would the annuity company gain ? 4. If he were to live 43 years, how much would the company lose ? 5. A man purchases an annuity for life, on the supposition that he shall live 45 years, for $15000, and is allowed 4 per cent. interest. How much must he draw annually that the whole may be exhausted ? 6. A man has property to the amount of $35000, which yields him an income of 5 per cent. His annual expenses are $5000. How long will his property last him ? 7. The number of slaves in the United States in 1810 was 1,191,000, and in 1820 the number was 1,531,000. What is the number at present, 1925, allowing the rate of increase to be the same ? 8. There is a society, established in the United States for the purpose of colonizing the free people of colour. Suppose the slaves to be emancipated as fast as this society can transport them away ; how many must be sent away annually, that the number may be neither increased nor diminished ? 9. How many must be sent away annually that the country may be cleared in 100 years ? 10. If the colonization is not commenced till the year 1840, supposing the rate of increase to remain the same as from 1810 to 1820, how many must then be sent away annually, that the number remaining may continue the same? 11. How many must then be sent away annually, that the country may be cleared of them in 100 years ? Miscellaneous Examples. 1. An express set out to travel 240 miles in 4 days, but in consequence of the badness of the roads he found that he must go 5 miles the second day, 9 the third, and 14 the fourth, less than the first. How many miles must he travel each day? 2. Two workmen received the same sum for their labour ; but if one had received 27 shillings more and the other 19 shillings less, then one would have received just three times as much as the other. What did they receive? 3. Two persons, A and B worked together, A worked 15 and B 18 days, and they received equal sums for their work. But if A had worked 174 and B 14 days, then A would have received 35 shillings more than B. What was the daily wages of each ? 4. Two merchants entered into a speculation, by which one gained 54 dollars more than the other. The whole gain was 49 dollars less than three times the gain of the less What were the gains ? 5. A man bought a piece of cloth for a certain sum, and on measuring it, found that it cost him 8 dollars, but if there had been 4 yards more, it would have cost him only $7 per yard. How many yards were there? 6. Divide the number 46 into two such parts, that one of them being divided by 7, and the other by 3, the quotients may together be equal to 10. 7. A farm of 864 acres is divided between 3 persons. C has as many acres as A and B together; and the portions of A and B are in the proportion of 5 to 11. How many acres had each ? 8. There are two numbers in the proportion of fto , the first of which being increased by 4 and the second by 6, they will be in the proportion of toʻ1. What are the numbers? 9. A farmer has a stack of hay, from which he sells a quantity, which is to the quantity remaining in the proportion of 4 to 5. He then uses 15 loads, and finds that he has a quantity left, which is to the quantity sold as 1 to 2. How many loads did the stack at first contain ? 10. There are 3 pieces of cloth, whose lengths are in the proportion of 3, 5, and 7; and 8 yards being cut off from each, the whole quantity is diminished in the proportion of 15 to 11. What was the length of each piece at first? 11. The number of days that 4 workmen were employed were severally as the numbers 4, 5, 6,7; their wages were the ages ? 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 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 men, 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 ? |