117. 12. What is the worth of a freehold estate, of which the yearly rent is $60, allowing to the purchaser 6 per cent.? In this case, the annuity continues forever, and the estate is evidently worth a sum, of which the yearly interest is equal to the yearly rent of the estate. The principal multiplied by the rate gives the interest; therefore, the interest divided by the rate will give the principal; 60'06 1000. Ans. $1000. = Hence, to find the present worth of an annuity, continuing forever,-Divide the annuity by the rate per cent., and the quotient will be the present worth. Note. The worth will be the same, whether we reckon simple or compound interest; for, since a year's interest of the price is the annuity, the profits arising from that price can neither be more nor less than the profits arising from the annuity, whether they be employed at simple or compound in terest. 13. What is the worth of $100 annuity, to continue forever, allowing to the purchaser 4 per cent.? 5 per cent.? per cent. ? 8 per cent.? 20 per cent. ? allowing 10 per cent.? - 15 Ans. to last, $500. 14. Suppose a freehold estate of $60 per annum, to commence 2 years hence, be put on sale; what is its value, allowing the purchaser 6 per cent. ? Its present worth is a sum which, at 6 per cent. compound interest, would, in 2 years, produce an amount equal to the worth of the estate if entered on immediately. 60 $1000 the worth, if entered on immediately, '06 and $1000 1'062 $889'996, the present worth. The same result may be obtained by subtracting from the worth of the estate, to commence immediately, the present worth of the annuity 60, for 2 years, the time of REVERSION. Thus, by the table, the present worth of $1 for 2 years is 1'83339 X 60 = 110'0034 present worth of $60 for 2 years, and $1000 $110'0034 = $889'9966, Ans. as before. 15. What is the present worth of a perpetual annuity of $100, to commence 6 years hence, allowing the purchaser 5 per cent. compound interest? what, if 8 years in re version? 30 years? 10 years? 4 years? 15 years? Ans. to last, $462'755. The foregoing examples, in compound interest, have been confined to yearly payments; if the payments are half yearly, we may take half the principal or annuity, half the rate per cent., and twice the number of years, and work as before, and so for any other part of a year. QUESTIONS. the 1. What is a geometrical progression or series? 2. What is the ratio? 3. When the first term, the ratio, and the number of terms, are given, how do you find the last term? 4. When the extremes and ratio are given, how do you find the sum of all the terms? 5. When the first term, the ratio, and the number of terms, are given, how do you find the amount of the series? 6. When the ratio is a fraction, how do you proceed? 7. What is compound interest? 8. How does it appear that the amounts, arising by compound interest, form a geometrical series? 9. What is the ratio, in compound interest? the number of terms? first term? the lust term? 10. When the rate, the time, and the principal, are given, how do you find the amount? 11. When A. R. and T. are given, how do you find P. 12. When A. P. and T. are given, how do you find R.? 13. When A, P. and R. are given, how do you find T.? 14. What is an annuity? 15. When are annuities said to be in arrears? 16. What is the amount? 17. In a geometrical series, to what is the amount of an annuity equivalent? 18. How do you find the amount of an annuity, at compound_interest? 19. What is the present worth of an annuity? how computed at compound interest? how found by the table? 20. What is understood by the term reversion? 21. How do you find the present worth of an annuity, taken in reversion? by the table? 22. How do you find the present worth of a freehold estate, or a perpetual annuity? same taken in reversion? by the table? the PERMUTATION. ¶ 118. Permutation is the method of finding how many different ways the order of any number of things may he varied or changed. 1. Four gentlemen agreed to dine together so long as they could sit, every day, in a different order or position; how many days did they dine together? Had there been but two of them, a and b, they could sit only in 2 times 1 (1 × 2 = 2) different positions, thus, a b, and b a. Had there been three, a, b, and c, they could sit in 1 × 2 × 36 different positions; for, beginning the order with a, there will be 2 positions, viz. a b c, and a cb; next, beginning with b, there will be 2 positions, ba c, and bea; lastly, beginning with c, we have c a b, and c ba, that is, in all, 1 X 2 X 3 6 different positions. In the same manner, if there be four, the different positions will De 1 X 2 X 3 X 4 = 24. Ans. 24 days. Hence, to find the number of different changes or permutations, of which any number of different things are capable,Multiply continually together all the terms of the natural series of numbers, from 1 up to the given number, and the last product will be the answer. 2. How many variations may there be in the position of the nine digits ? Ans. 362880. 3. A man bought 25 cows, agreeing to pay for them 1 cent for every different order in which they could all be placed; how much did the cows cost him? Ans. $155112100433309859840000. 4. Christ Church, in Boston, has 8 bells; how many changes may be rung upon them? Ans. 40320. MISCELLANEOUS EXAMPLES. ¶ 119. 1. 4+6 × 7-1=60. A line, or vinculum, drawn over several numbers, signifies, that the numbers under it are to be taken jointly, or as one whole number. 2.9-8+4x8+4-6 how many? 3.7+4-2+3+40 x 5 = how many? Ans. 30. Ans. 230 Ans. 3 5. There are two numbers; the greater is 25 times 78, and their difference is 9 times 15; their sum and product are required. Ans. 3765 is their sum; 3539250 their product 6. What is the difference between thrice five and thirty, and thrice thirty-five? 35 X 3-5 X 3 +30= 7. What is the difference between six dozen dozen, and half a dozen dozen? : 60, Ans. Ans. 792. 8. What number divided by 7 will make 6488? 9. What number multiplied by 6 will make 2058 ? 10. A gentleman went to sea at 17 years of age; 8 years after he had a son born, who died at the age of 35; after whom the father lived twice 20 years; how old was the father at his death? Ans. 100 years. 11. What number is that, which being multiplied by 15 ÷ 15, Ans. the product will be ? 12. What decimal is that, which being multiplied by 15, the product will be "75? 75 ÷ 15'05, Ans. 13. What is the decimal equivalent to? Ans. '0285714. 14. What fraction is that, to which if you add 2, the sum will be ? 15. What number is that, from which if you take g, the remainder will be ? Ans. 18. Ans. 18 16. What number is that, which being divided by, the quotient will be 21 ? Ans. 15. 17. What number is that, which multiplied by produces? 18. What number is that, from which if you itself, the remainder will be 12? 19. What number is that, to which if you add itself, the whole will be 20? 20. What number is that, of which 9 is the Ans. take of Ans. 20. of § of Ais. 12. part? Ans. 13. 21. A farmer carried a load of produce to market: he sold 780 lbs. of pork, at 6 cents per lb. ; 250 lbs. of cheese, at 8 cents per lb.; 154 lbs. of butter, at 15 cents per lb. : in pay he received 60 lbs. of sugar, at 10 cents per lb.; 15 gallons of molasses, at 42 cents per gallon; barrel of mackerel, at $375; 4 bushels of salt, at $125 per bushel; and the balance in money: how much money did he receive? Ans. $68'85. 22. A farmer carried his grain to market, and sold 75 bushels of wheat, at $1'45 per bushel. 64 ... $ '95 In exchange he received sundry articles : 8 pieces of cloth, each containing 31 yds., at 2 quintals of fish, 8 hhds. of salt, ... and the balance in money. $175 per yd. $2'30 per quin. Ans. $38'80. How much money did he receive? 23. A man exchanges 760 gallons of molasses, at 37 cents per gallon, for 663 cwt. of cheese, at $4 per cwt.; how much will be the balance in his favour? Ans. $19. 24. Bought 84 yards of cloth, at $125 per yard; how much did it come to? How many bushels of wheat, a $150 per bushel, will it take to pay for it? Ans. to the last, 70 bushels. 25. A man sold 342 pounds of beef, at 6 cents per pound, and received his pay in molasses, at 371 cents per gallon; how many gallons did he receive? Ans. 5472 gallons. 26. A man exchanged 70 bushels of rye, at $92 per bushel, for 40 bushels of wheat, at $1'37 per bushel, and received the balance in oats, at $40 per bushel; how many bushels of oats did he receive? Ans. 231. 27. How many bushels of potatoes, at 1 s. 6 d. per bushel, must be given for 32 bushels of barley, at 2 s. 6 d. per bushel? Ans. 53 bushels. 28. How much salt, at $150 per bushel, must be given in exchange for 15 bushels of oats, at 2 s. 3 d. per bushel? Note. It will be recollected that, when the price and cost are given, to find the quantity, they must both be reduced to the same denomination before dividing. Ans. 3 bushels. 29. How much wine, at $275 per gallon, must be given in exchange for 40 yards of cloth, at 7 s. 6 d. per yard? Ans. 18 gallons. |