many dollars, for the same time and at the same rate, as $.24 is contained times in $87.42. Dividing, we obtain $364.25, the required principal. Hence the RULE. Divide the given interest by the interest of $1 for the given time at the given rate. EXAMPLES FOR PRACTICE. 1. What sum of money, invested at 63 per cent., will produce $279.825 in 1 yr. 6 mo.? Ans. $2870. 2. What. sum will produce $63.75 interest in 6 mo. 24 da. at 72 per cent.? 11500. 3. What sum will produce $12 interest in 10 days at 10 per cent.? Ans. $4500. 4. What sum must be invested in real estate paying 121 per cent. profit in rents, to give an income of $3125? 25000 5. What is the value of a house and lot that pays a profit of 93 per cent. by renting it at $30 per month? 3782474 6. What sum of money, put at interest 6 yr. 5 mo. 11 da、, at 7 per cent., will gain $3159.14? Ans. $7000. 7. What sum of money will produce $69.67 in 2 yr. 9 mo. at 6% compound interest? Ans. $400. 8. What principal at 6 % compound interest will produce $124.1624 in 1 yr. 6 mo. 15 da.? Ans. $1314.583. PROBLEM II. 553. Given, the time, rate per cent., and amount, to find the principal. 1. What sum of money in 2 years 6 months, at 7 will amount to $136.535? OPERATION. $1.175, amount of $1 for 2 yr. 6 mo. $136.535 1.175 $116.20, Ans. per cent., ANALYSIS. Since $1.175 is the amount of $1 for 2 years 6 months, at 7 per dollars, for the same times in $136.535. cent., $136.535 must be the amount of as many RULE. Divide the given amount by the amount of $1 for the given time at the given rate. EXAMPLES FOR PRACTICE. 1. What principal in 2 yr. 3 mo. 10 da., at 5 per cent., will amount to $1893 611 ? Ans. $1700. 2. A note which had run 3 yr. 5 mo. 12 da. amounted to $681.448, at 6 per cent.; how much was the face of the note 3. What sum put at interest at 3 per cent., for 10 will amount to $15660? yr. 2 mo 11550.09 4. What is the interest of that sum for 2 yr. 8 mo. 29 da., at 7 per cent., which at the same time and rate, will amount to $1568.97 ? Ans. $253.057 +. 5. What is the interest of that sum for 243 days at 8 per cent., which at the same time and rate, will amount to $11119.70? 6. What principal in 4 years at 6 per cent. compound interest, will amount to $8644.62? Ans. $6847.34. 7. What sum put at compound interest will amount to $26772.96, in 10 5 yr. mo., at 6 per cent. ? PROBLEM III. Ans. $14585.24. 554. Given, the principal, time, and interest, to find the rate per cent. 1. I received $315 for 3 years' interest on a mortgage of $1500; what was the rate per cent.? OPERATION. $15.00 $45.00, int. for 3 yr. at 1 %. $315 - $45 7 %, Ans. as many times 1 per cent. as $45 is contained ANALYSIS. Since $45 is the interest on the mortgage for 3 years at 1 per cent., $315 must be the interest on the mortgage for the same time, at times in $315. Divid ing, and we obtain 7, the required rate per cent. Hence the RULE. Divide the given interest by the interest on the principal for the given time at 1 per cent. EXAMPLES FOR PRACTICE. 1. If I loan $750 at simple interest, and at the end of 1 mo. receive $796.873, what is the rate per cent.? yr. 3 Ans. 5.. 2 If I pay $10.58 for the use of $1700, 28 days, what is the rate of interest? Ans. 8 per cent. f. 3. Borrowed $600, and at the end of 9 yr. 6 mo. returned $856.50; what was the rate per cent.? 4. A man invests $7266.28, which gives him an annual income of $744.793; what rate of interest does he receive? 10 5. If C buys stock at 30 per cent. discount, and every 6 months receives a dividend of 4 per cent., what annual rate of interest does he receive? Ans. 11 per cent. 6. At what rate per annum of simple interest will any sum of money double itself in 4, 6, 8, and 10 years, respectively? 7. At what rate per annum of simple interest will any sum triple itself in 2, 5, 7, 12, and 20 years, respectively? 8. A house that rents for $760.50 per annum, cost $7800: what % does it pay on the investment? Ans. 9 per cent. 9. I invest $35680 in a business that pays me a profit of $223 a month; what annual rate of interest do I receive? Ans. 7%. PROBLEM IV. 555. Given, the principal, interest, and rate, to find the time. 1. In what time will $924 gain $151.536, at 6 per cent.? $924 OPERATION. $55.44, int. of $960 for 1 yr. at 6 %. 2.73 yr. 2 yr. 8 mo. 24 da., Ans. ANALYSIS. Since $55.44 is the interest of $924 for 1 year at 6 per cent., $151,536 must be the interest of the same sum, at the same rate per cent., for as many years as $55.44 is contained times in $151.536, which is 2.73 times. Reducing the mixed decimal to its equivalent compound number, and we have 2 years 8 months 24 days, the required time. Hence the RULE. Divide the given interest by the interest on the principal for 1 year; the quotient will be the required time in years and decimals. EXAMPLES FOR PRACTICE. 1. In what time will $273.51 amount to $312.864, at 7 per cent.? Ans. 2 yr. 20 da. 2. How long must $650.82 be on interest to amount to $761.44, per cent.? Ans. 3 yr. 4 mo. 24 da. at 5 Ans. 3. How long will it take any sum of money to double itself by simple interest at 3, 41, 6, 7, and 10 per cent.? How long to quadruple itself? (To double itself at 3 %, 33 vr. To quadruple itself at 3 %. 100 yr. 4. In what time will $9750 produce $780 interest, at 2 per cent. a month? 4 mo. { ANALYSIS. $1171.353÷1000-$1.171353, the amount of $1 for the required time. From the table, $1, in 2 years, will amount to $1.1236; hence $1.171353÷$1.1236=$.047753, the interest which must accrue on $1.1236 for the fraction of a year; and $1.1236×.06=$.067416; $.047753÷$.067416=.7083 yr.=8 mo. 15 da. Ans. 2 yr. 8 mo. 15 da. 5. In what time will $1000 draw $1171.353 at 6 per cent. compound interest? Ans. 2 yr. 8 mo 15 da. 6. In what time will $333 amount to $376.75 at 5 per cent compound interest, payable semi-annually? 2 7. In what time will any sum double itself at 6 % compound interest? At 7 %? Ans. to last, 10 yr. 2 mo. 28 da. ..FIRST 11-10-21 DISCOUNT. 56. Discount is an abatement or allowance made for the payment of a debt before it is due. 557. The Present Worth of a debt, payable at a future time without interest, is such a sum as, being put at legal interest, will amount to the given debt when it becomes due. 1. What is the present worth and what the discount of $642.12 to be paid 4 yr. 9 mo. 27 da. hence, money being worth 7 per rent.? in $642.12. Dividing, and we obtain $480 for the present worth, and subtracting this sum from the given sum, we have $162.12, the discount. Hence the following RULE. I Divide the given sum or debt by the amount of $1 for the given rate and time; the quotient will be the present worth of the debt. II. Subtract the present worth from the given sum or debt; the remainder will be the discount. NOTES.-1. The terms present worth, discount, and debt, are equivalent to principal, interest, and amount. Hence, when the time, rate per cent., and amount are given, the principal may be found by Prob. II, (553); and the interest by subtracting the principal from the amount. 2. When payments are to be made at different times without interest, find the present worth of each payment separately. Their sum will be the present worth of the several payments, and this sum subtracted from the sum of the several payments will leave the total discount. EXAMPLES FOR PRACTICE. 1. What is the present worth of a debt of $385.314, to be paid in 5 mo. 15 da., at 6 %? Ans. $375. 2. How much should be discounted for the present payment of a note for $429.986, due in 1 yr. 6 mo. 1 da., money being worth 51% ? Ans. $32.826. 3. Bought a farm for $2064.12 ready money, and sold it again for $3665.20, payable in 1 yr. 6 mo. How much would be gained in ready money, discounting at the rate of 8 %? 3081 4. A man bought a flouring mill for $25000 cash, or for $12000 payable in 6 mo. and $15000 payable in 1 yr. 3 mo. He accepted the latter offer; did he gain or lose, and how much, money being worth to him 10 per cent.? Ans. Gained $238.10. 5. B bought a house and lot April 1, 1860, for which he was to pay $1470 on the fourth day of the following September, and |