Compound Interest. 179. What will be the interest of $40 for 3 years, at 6 per cent., the interest being added to the principal at the end of each year? The interest of 40 dollars for 1 year is (40X.06) $2.40, and $2.40+ 40.842.40, the principal for the second year, the interest of which is (42.40.06 ) $2.544 for the second year, and $2.514+42.40 $44.944, the principal for the third year, the interest of which is (44.944.06=) $2.696, and $2.696+44.944—$47.64, the amount of principal and interest at the end of three years, from which subtracting 40 dollars, the first principal, we have (47.64 40.) $7.64 for the interest of 40 dollars for years. Interest computed upon interest, as above, is called Compound Interest. 180. COMPOUND INTEREST is that which arises from making the interest a part of the principal at the end of each year, or stated time for the interest to become due. RULE. Find the amount of the given principal for the first year, or up to the first stated time for the interest to become due, by simple interest, and make the amount the principal for the next year, or stated period; and so on to the last. From the last amount subtract the given principal, and the remainder will be the compound interest required. QUESTIONS FOR PRACTICE. 2. What is the compound interest of $100 for 4 years, at 6 per cent.? Ans. $26.247. 3. What is the compound interest of $200 for 1 year, at 6 per cent., due every four months?. Ans. $12.241. 4. What is the amount of $236 at 6 per cent., compound interest, for 3 years, 5 months, and 6 days? Ans. $288.387. 5. What is the amount of $150 at 6 per cent., compound interest, for 2 years, the, interest becoming due at the end of every 6 months? Ans. $168.826. 6. What is the compound interest of $768 for 4 years, at cent.? Ans. $201.58. 7. What is the compound interest of $560 for 3 years and 6 months, at 6 per cent.? Ans. $126.977. 3. Discount. 181. A holds a note against B for $213, payable in one year and six months without interest, which he wishes to turn out to B in payment for a farm; what is the present worth of the note, supposing the use of money to be worth 6 per cent. per annum? As the amount of 1 dollar for 1 year and 6 months, at 6 per cent. is $1.09, 1 dollar is evidently the present worth of $1.09 due 1 year and 6 months hence, without interest; because, if 1 dollar be put to interest at the above rate, at the end of 1 year and 6 months, the amount will be just sufficient to pay the $1.09. Now, as one dollar is the present worth of $1.09, due 18 months hence, the present worth of any other sum, at the same rate and for the same time, is evidently as many dollars as the number of times that sum contains $1.09. Hence to find the present worth of $218, due 18 months hence, we divide $218 by $1.09, and the quotient (218-1.09 ) $200 is the present worth. If we subtract the present worth from the amount of the note, the difference, (218—200—) $18, is called the discount. The interest of the given sum for the above time and rate, would have been $19,62, greater than the discount by $1.62. DISCOUNT 182. Is an allowance made for the payment of money before it is due, or so much per cent. to be deducted from a given sum. The present worth of a sum of money due some time hence, and not on interest, is such a sum as would, if put to interest, at a given rate, at the end of the given time, just amount to the sum then due. RULE. 183. Divide the given sum by the amount of 1 dollar for the given time and rate, and the quotient will be its present worth. Subtract the present worth from the given sum, and the remainder will be the discount. QUESTIONS FOR PRACTICE. 2. What is the present worth | at 6 per cent. per ann.: how of $125, due 3 years hence, much ready money must they discounting at the rate of 6 pay? Ans. $1047.047. per cent. per annum? Ans. $105.9321. 3. What is the present worth of $376.25, due at the end of 1 year and 6 mos., discounting at 5 per cent.? Ans. $350. 4. A minister settled with a salary of $300 a year: wishing to build a house, his parishioners agreed to pay him 4 years salary in advance, discounting 5. What is the present worth of $150, payable in 3 months, discount 5 per cent. ? Ans. $148.148. 6. What is the discount upon $560, due 9 months hence, at 8 per cent.? Ans. $31.698.8. 7. What is the dis $50, due 2 years her per cent.? Ans 4. Loss and Gain. 184. If I buy a horse for $50, and sell it again for $56, what do I gain per cent.? Subtracting 50 dollars from 56 dollars, we find that 50 dollars gains 6 dollars, and dividing 6 dollars by 50 dollars, we find $0.12 to be the gain on $1, or 12 cents on 100 cents, or $12 on $100, or 12 per cent. Hence 185. To know what is gained or lost per cent. RULE. Find the gain or loss on the given quantity by subtraction. Divide this gain or loss by the price of the given quantity, and the quotient will be the gain or loss per cent. QUESTIONS FOR PRACTICE. 2. If I buy cloth for $1.25 | a yard, and sell it again for $1.30, what do I gain per ct.? 1.25).0500(0.04 per cent. 500 Ans.* 3. If I buy salt for 84 cents a bushel, and sell it for $1.12 a bushel, what do I gain per cent.? Ans. $0.33 per cent. 4. If I buy cloth for $1.25 a yard, and sell it for $1.37 a yard, what do I gain per cent.? Ans. $0.10 per cent. 5. If I buy cloth at $1.02 a yard, and sell it at $0.90, what do I lose per cent.? Ans. $0.1113. 6. If corn be bought for $0.75, and sold for $0.80 a bushel, what is gained per ct.? Ans. $0.06. These answers properly express the number of cents, loss or gain, on the dollar. If the decimal point be taken away, they will express the number of dollars on the $100. 186. If I buy tea for 75 cents a pound, how must I sell it to gain 4 per cent.? $0.75 at 4 per cent. is (.75X.04) $0.03, and 75+.03 $0.78, the selling price. The method in this case is precisely the same as that for interest for one year (160). If, instead of gaining, I wish to lose 4 per cent., the .03 must be subtracted from .75, leaving .72 for the selling price. Hence 187. To know how a commodity must be sold to gain or lose so much per cent. RULE.-Multiply the price it cost by the rate per cent., and the product added to, or subtracted from, this price, will be the gaining or losing price. QUESTIONS FOR PRACTICE. 2. If I buy cloth for $0.75, how must I sell it to gain 94 Ans. $0.8211. per cent.? 3. If I buy corn for $0.80 a bushel, how must I sell it in order to lose 15 per cent.? Ans. $0.68. 4. Bought 40 gals. of rum at 75 cents a gallon, of which 10 gallons leaked out: how must I sell the remainder, in order to gain 123 per cent, on the prime cost? Ans. $1.125 per gallon. 5. Equation of Payments. 188. A owes B 5 dollars, due in 3 months, and 10 dollars due in 9 months, but wishes to pay the whole at once; in what time ought he to pay it? $5, due in 3 months $1, due in 15 months, and $10, due in 9 months $1, due in 90 months; then (5+10=) $15, due $5 in 3 months, and 10 in months $1 due in (15+90) 105 months. Hence, A might keep $1, 105 months, or $15, of 105 months, or 105-7 months. This method of considering the subject supposes that there is just as much gained by keeping a debt a certain time after it is due, as is lost by paying it an equal length of time before it is due. But this is not exactly true; for by keeping a debt unpaid after it is due, we gain the interest of it for that time; but by paying it before it is due, we lose only the discount, which has been shown to be somewhat less than the interest (181). The following rule, founded on the analysis of the first example, will, however, be sufficiently correct for practical purposes. 189. RULE.-Multiply each of the payments by the time in which it is due, and divide the sum of the products by the sum of the payments; the quotient will be the equated time of payment. 6. What is the present worth of 426 dollars, payable in 4 years and 12 da. at 5 per cent.? Ans. $354.409. 7. What is the present worth of 960 dollars, payable as follows, viz. in 3 months, in 6 months, and the rest in 9 months, discount to be made at 6 per cent.? Ans. $936.70. 8. A buys a quantity of rice for $179.56; for what must he sell it to gain 11 per cent.? Ans. $199.311. NOTE. It will be observed that the result obtained by the second method differs very materially from the others. But that result is evidently erroneous and unjust; for the debtor being under no obligation to make payments before the time specified in the note, he might have let out these payments upon interest till that time, and then the amount of these taken from the amount of the principal, would leave the balance justly due, and which would be the same as that found by method III. Hence, in computing interest on notes, bonds, &c. the conditions of the contract should always be taken into consideration. The second method is applicable to notes which are payable on demand, especially after a demand of payment has been made, and also to other contracts after the specified time of payment is past. REVIEW. 1. What is meant by the term per cent.?-by per annum? 2. What is meant by Interest ?by the principle ?-by the rate per cent. ?-by the amount? 3. Of how many kinds is Interest? 4. How is the rate per cent. expressed? What do decimals in the rate below hundreds express? Is rate established by law? What is it in New England in New York? 5. What is Simple Interest? 6. How would you find the interest on any sum for one year? For more years than one? Repeat the rule for the first method. 7. How would you proceed, if the principal were in English Money? 8. If interest be allowed at 12 per cent, what would be the monthly rate? How then would you cast the interest on a given sum for a given time at 12 per cent.? 9. What part of 12 per cent. is 6 per cent.? What then would be the monthly rate at 6 per cent.? 10. What is the second method of casting interest at 6 per cent.? What is done with the odd days, if any less than 6? Having found by this method the interest at 6 per cent., how may it be found for any other per cent.? What is the rule which is to be observed in all cases for pointing? (122) 11. The time, rate, and amount being given, how would you find the principal? 12. The time, rate, and interest being given, how would you find the principal ? |