What per cent. of the said sum is remaining in the bank ? What per cent., predicating it upon the first-mentioned balance, have I drawn? What per cent. have I drawn, predicating it upon what I now have in the bank? What amount of money must I deposit to make good 621 per cent. of the aforementioned sum? d=r (a+m) + b'+b" — (a + m)=r (a+m)—d"= To find the Principal or Basis. EXAMPLES. $22.96. Ans. The percentage being 250, and the rate .06, what is the principal? a=b÷r=100b÷p=250.0625,000—6=4,1663. Ans. A tax at the rate of 5 of 1 per cent. on the valuation was $27.50. What was the valuation? Sold 120 barrels of flour, which amounted to 12 per cent. of a certain consignment. The consignment consisted of how many barrels ? 120 0.121,000. Ans. 216 bushels is more by 8 per cent., or 8 per cent. more, than what number of bushels? 8 per cent. more than what number is equal to 216? What number, plus 8 per cent. of it, will make 216? a=s÷(1+r) = = 2161.08- 200. Ans. 200 lbs. is less by 8 per cent., or 8 per cent. less, than what num ber of lbs.? 8 per cent. less than what number is 200? What number, minus 8 per cent. of it, is equal to 200 ? a=d÷(1 − r) = 200 ÷ (1— .08)=217. Ans. · . 217 — 217.08=200=a—b=d=a(1 —r). To a quantity of silver, a quantity of copper equal to 20 per cent. of the silver is to be added, and the mass is to weigh 22 ounces. What weight of silver is required? a=s÷(1+r)=221.218 ounces. Ans. What weight of copper is required? To a quantity of copper, a quantity of nickel equal to 62 per cent. of the copper, a quantity of zinc equal to 333 per cent. of the copper, and a quantity of lead equal to 5 per cent. of the copper, are to be added; and the whole is to weigh 40 pounds. weight of each constituent of the alloy is required. The T=time in months and decimal parts of a month; t=time in days; P= principal; r=rate per cent., expressed decimally; i=interest. EXAMPLE. A promissory note, made April 27, 1864, for t= 12 i r= 365 i = Pr $825,25 and interest at 6 per cent., matured Oct. 6, 1865: what was the interest? Oct. is 10th month. d. 6 27 Time from April 27 to Oct. 6 (one of the dates always included) 162 days, which, added to the 365 days in the year preceding =527 days. NOTE. One day's interest at least is generally lost by computing the time in years and 9 months, or months, instead of days. 825.25 X 17.3 X .0612=$71.38. 825.25 527 X.06365$71.49. Ans. Ans. To find a constant divisor, k, for any given rate per cent. When the time is taken in months, k12r. When the time is taken in days, k—365÷r; thus, PX t =Interest. 6083 PX t Interest, &c. 5214 When the RATE is 7 per cent. EXAMPLE. per cent. EXAMPLE. in 3 years? EXAMPLE. Required the interest on $750 for 93 days, at 7 750 X 935214 $13.38. Ans. - What is the rate per cent. when $450 gains $94 450: 100 :: 94.5; 3x= 7 per cent. Ans. 94.53 X 450.07. Ans. In what time will $125 at 6 per cent. gain $18§? 6: 100 :: 18.75: 125 × x = 21 years. Ans. 18.75125 .06 = 21 years. Ans. EXAMPLE.What principal at 5 per cent. interest will gain $167 in 18 months? 5 100 16.875 1.5 X x = $225. Ans. 16.875 X 1218 X.05 $225. Ans. When partial payments have been made. RULE. - Find the amount (sum of the principal and interest) up to the time of the first payment, and deduct the payment therefrom; then find the interest on the remainder up to the next payment, add it to the remainder, or new principal, and from the sum subtract the next payment; and so on for all the payments; then find the amount up to the time of final payment for the final amount. COMPOUND INTEREST. If we calculate the interest on a debt for one year, and then on the same debt for another year, and again on the same debt for still another year, the sum will be the simple interest on the debt for three years. But, on the contrary, if we calculate the interest on the debt for one year, and then on the amount (sum of the principal and interest) for the next year, and then on the second amount for the third year, the sum of the interest so calculated will be the compound interest, or yearly compound interest, on the debt for three years; equal to the simple interest on the debt for three years, plus the yearly compound interest on the first year's interest for two years, plus the simple interest on the second year's interest for one year. So, if we divide the time into shorter periods than a year, and proceed for the interest as last suggested, the interest will be compound. Thus we have half-yearly compound interest, or compound interest semi-annually, quarteryearly compound interest, or compound interest quarterly, &c. This method of computing interest is predicated upon the natural idea, that interest, when it becomes due by stipulation and is withheld, commences to draw interest, and continues at use to the holder, at the same rate as the principal, until it is paid, like other over-due demands; and that the interest so made matures and becomes due as often, and at the same periods, as that on the principal. It will be perceived by the foregoing that the working-time in compound interest is the interval between the stipulated payments of the interest, or between one stipulated payment of the interest and that of another; and that the working-rate is pro rata to the rate per annum. Thus the amount of $100 at semi-annual compound interest for years, at 6 per cent. per annum, is 100 X (1.03) 4 = $112.550881=$112.55, or If we let P principal or debt at interest, rworking-rate of interest, n number of intervals into which the whole time is divided for the payment of interest, or number of consecutive intervals for the payment of interest that have transpired without a payment having been made, icompound interest, A P(1+r)"; P= A A A (1+r) air = ·= (1+r)"; i = A— P. P EXAMPLE. What is the compound interest, or yearly compound interest, on $100 for 14 years, at 6 per cent. a year? 100 × 1.06 X 1.03109.18-100 $9.18. Ans. EXAMPLE.What is the amount of $560.46, at 7 per cent. compound interest per year, for 6 years and 57 days? |