5. If merchandize is sold at 5 per ct. deduction on the cost, to the amount of $1,380.00 when money will bring 5 per ct.; what is the total loss, supposing it has been 5 months on hand, and 3 months' credit is given ? 6. A merchant sold 3 pieces of broadcloth, each piece containing 27 yards, at 7 dollars a yard, making an advance of 12 per ct. on the cost; it had been 3 months on hand, and 2 months credit was given : 7 pipes of wine at $4.50 per gallon, at an advance of 18 per ct. on the cost, which had been 7 months on hand, and for which he gave 3 months credit; and 7 bales of cotton at 11 cents a pound, each bale containing 230 pounds, which had been on hand 1 mo. 15 days, at an advance of 20 per ct. on the cost, and giving 6 months credit. What was the whole amount of his profit, and his profit on each article, supposing that money will command 4 per ct.? Also, what was his real gain per ct. on each article, and his real gain per cent. on the whole? Ans. Whole gain $566.0493213313-On each art. $51.93211-$487.660355.-$26.456187.--Rates.10, -148-17-Average rate .14.196445181680 29546723858 18 39 $ LXXXVI. DUTIES. When a duty is said to be AD VALOREM, it is meant that it is at a certain rate on the value of the articles. The term is used to distinguish this class of duties from those imposed on the quantity; as, a duty upon the gallon, pound, barrel, cwt., ton, &c. A written account of articles, sent to a purchaser, factor, or consignee, with the prices and charges annexed, is called an INVOICE. In computing duties, ad valorem, (or ad val. as it is commonly written,) it is nsual in custom houses to add one tenth to the invoice value, before casting the duty. This makes the real duty one tenth greater than the nominal duty. It will be equally well to make the rate one tenth greater, instead of increasing the invoice. 1. Find the duty on a quantity of tea, of which the invoice is $215.17, at 50 pr. ct. Ans. $118.3435-$118.343. In this example we may add, as directed above, one tenth of 215.17 to 215.17. Thus, 215.17+21.517=236.687. Then, 236.687 X50-$118.3435. Or we may add to the rate .50, one tenth of itself =.05: thus, .50+.05.55. Then, 215.17×.55 $118.3435, as before. 2. Find the duty on a quantity of hemp, at 131 pr. ct, of which, the invoice is $654.59. The second of the above modes is recommended. Another might be used-viz.: to find, first, the duty on the invoice at the given rate, and add to it one tenth of itself. Thus, 654.59×13=$88.36965. Ans. $97.206615. 3. What is the duty on a quantity of books, of which the invoice is $1,670.33, at 20 pr. ct.? A. $367.4726. 4. Find the duty on a quantity of wine, of which the invoice is $2,964.666, at 45 pr. ct. A. $1,467.50967. 5. Find the duty on a quantity of merchandize, of which the in. voice is $6,954.73, at 18 pr. ct. 6. Find the duty on a quantity of goods, of which the invoice is $7,458.133, at 15 pr. ct. § LXXXVII. EQUATION OF PAYMENTS. It is sometimes required to know at what time several debts, which fall due at dif. ferent times, may be paid, so that neither creditor nor debtor shall lose. In this case, it is plain, that those due latest, must be paid before they are due, while those due earliest, may be kept after they are due. MENTAL EXERCISES. 1. How long will $1.00 be in gaining as much as $3.00 gains in one year? 2. How long will it take $1.00 to gain as much as $5.00 in 6 mo. ? 3. How long will $5.00 be in gaining as much as $1.00 gains in 50 yrs. ? 4. How many months will it take $7.00 to gain as much as $1.00 in 35 mo. ? 5. How long will $2.00 be in gaining as much as $3.00 in 4 mo. ? Ans. $3.00 in 4 mo. would gain as much as $1.00 in 3X4=12 mo. $2.00 would be half as long as $1.00. 12÷2=6 mo. 6. How long will $3.00 be in gaining as much as $5.00 in 6 yrs. ? 7. How long will $4.00 be in gaining as much as $3.00 in 8 yrs. ? 8. How long will $9.00 be in gaining as much as $6.00 in 6 mo.? 9. How long will $6.00 be in gaining as much as $1.00 in 1 yr. ? 10. How long will $7.00 be in gaining as much as $1.00 in 3 yrs.? 11. How long will $4.00 be in gaining as much as $1.00 in 4 yrs. 4 mo.? 12. How long will $1.00 be in gaining as much as $9.00 in 9 yrs. ? Let the following be written. 13. How long will $1.00 be in gaining as mnch as $742.00 in 3 mo. A. 2,226 mo. 14. How long will $1.00 be in gaining as much as $895.00 in 5 mo. ? A. 4,475 mo. 15. How long will $1.00 be in gaining as much as $272.00 in 7 mo., and $336.00 in 6 mo. ? Ans. $272.00 for 7 mo.=$1.00 for 1,904 mo., and $336.00 for 6 mo. $1.00 for 2,016 mo. for 7 mo.+$336.00 for 6 mo. +2,016 mo.=3,920 mo. Ans. Then, $272.00 $1.00 for 1,904 mo. 16. How long will it take $1.00 to gain as much as $700.00 in 3 mo., $800.00 in 4 mo., $900.00 in 6 mo., and $500.00 in 12 mo. ? A. 16,700 mo. Then, it is plain that if I keep $1.00, 16,700 mo., I gain as much interest as I should by keeping the several sums in the example, the times specified. The amount of those sums is $2,900, for 700+800+900+500=2,900. Then, 17. How long may I keep $2,900.00 so as to gain as much interest as I gain on $1.00 in 16,700 mo.? I may keep it as long. A. 5 mo. 2223 d. Now, as $2,900.00 is the amount of all the sums in . ex. 16, if we consider them debts due at the times mentioned, it is plain that the whole might be paid in 5 mo. 222 d., without loss on either side. Hence, the two questions might have been expressed in one, as follows: 18. If I owe several debts, due as in example 16, at what time may I pay the whole together, without loss to me or to my creditor? A. In 5 mo. 2223 d. For $700 for 3 months is the same as $1 for 3X700=2,100 mo. Then, $2,900 for these several times, is the same as $1 for 16,700 mo. of the time=5 mo. 2222 d. Hence, to find the time when several payments, due at different times, may fairly be made at once, MULTIPLY EACH PAYMENT BY ITS TIME, AND DIVIDE THE SUM OF THE PRODUCTS BY THE SUM OF THE PAYMENTS. The time thus found is called the EQUAted time. When there are years and months, or years, months and days, it is best to reduce the months, or months and days, to a vulgar or decimal Fraction. A decimal is to be preferred. 19. Find the equated time for the following payments; viz. $100 due in 6 mo., $120 in 7 mo., and $160 in 10 mo. A. 8 mo. 20. Find the equated time for the following payments; viz. $50 in 2 mo., $100 in 5 mo., and $150 in 8 mo. A. 8 mo. 21. Find the equated time for paying $900 due in 4 mo., $700 due in 1 yr. 3 mo., $1,200 due in 2 yrs., and $600 due in 3 yrs. 6 mo. 22. Find the equated time for paying $900 due in 9 mo., $1,500 due in 6 yrs., $1,800 due in 7 yrs. 9 mo., and $1,000 due in 3 yrs. 8 mo., 12 d. It will be seen that the above rule supposes discount and interest to be equal, which is not true. But the error is too trifling to be regarded, in the common concerns of business. If any instance should occur, where the debts are very large, and the times very unequal, a rule for discovering the equated time may be found by an algebraic operation, but it would not be understood here, if given, and is therefore omitted. § LXXXVIII. COMPOUND INTEREST. 1. A owes B $600, on which he is under obligation to pay interest at the end of every year, until a final settlement is made. At the end of the first year, he finds it inconvenient to pay the interest, and B allows him to keep it, on condition that he will pay yearly interest on this likewise. For the second year, then, he is to pay interest on the original debt of $600, and also on the interest which he should have paid at the end of the first. What was the sum at interest the second year? A. $636. For 600X.06=36 the interest for 1st year, and 600+36= 636. 2. At the end of the second year, A is again unable to pay the interest, and the same agreement is made as before. What was the principal for the third year. A. $674.16. For 636.06=38.16 int. for 2nd year, and 636+38.16=674.16. 3. A settlement was made at the end of the 3rd year. What did A pay; and how much for interest? Ans. He paid $714.6096, for int. $114.6096. Interest, calculated thus on both principal and interest, is called COMPOUND INTEREST. Interest, calculated only on the principal, is called SIMPLE INTEREST. Though the only just interest is compound interest, it is not allowed by law. From the preceding examples, we derive the rule, I. I. FIND THE AMOUNT FOR THE FIRST YEAR, AS IN SIMPLE INTEREST, AND MAKE IT THE PRINCIPAL FOR THE SECOND; ON THIS, FIND THE AMOUNT FOR THE SECOND, AND MAKE IT THE PRINCIPAL FOR THE THIRD, AND SO ON, UP TO THE NUMBER OF YEARS GIVEN. THE LAST AMOUNT IS THE AMOUNT AT COMPOUND INTEREST. II. SUBTRACT THE PRINCIPAL FROM THE AMOUNT, and THE REMAIN DER WILL BE THE INTEREST. NOTE.-When there are months and days, calculate first for the years, and on the amount thus found, compute the interest for the months and days, which add to the amount previously obtained. 4. Required the amount of $100.00 for 3 years, at compound interest. A. $119.1016. 5. Required the compound interest on $300.00 for 4 years. A. $64.6519 In obtaining amounts, we may multiply the principal by the rate, with a unit added. (§ LXXIV. AND NOTE EX. 9. § LXXVI.) Hence, to find an amount at Compound interest, we may multiply the given principal, by the rate with a unit added, as many times successively as there are years. But this, (§ XII.) is the same as though we should multiply the principal, at once, by the product of all these successive multipliers; and as all the multipliers are alike, being the rate a unit, we have the following rule. II. I. ADD A UNIT TO THE RATE, AND MULTIPLY THE SUM BY ITSELF CONTINUALLY, UNTIL IT IS TAKEN AS A FACTOR, AS OFTEN AS THERE ARE YEARS. II. MULTIPLY THE PRINCIPAL BY THE LAST PRODUCT, AND THE RESULT WILL BE THE AMOUNT AT COMPOUND INTEREST. 6. Find the amount of $256, at compound interest, for 3 yrs. A. $304.900+ 7. Find the amount of $1,000.00 for 4 yrs. at 7 per ct. compound interest. A. $1,310.796+ 8. Find the amount of $425.00 for 4 yrs. at 5 per ct. compound A. $516.590+ interest. 9. Find the amount of $2,000.00 for 3 yrs. 6 mo. at compound interest. 10. Find the amount of $1,900.25 for 9 yrs. 7 mo. 7d. TABLE OF MULTIPLIERS, FOR FINDING COMPOUND INTEREST, ON ANY PRINCIPAL, FOR ANY NUMBER OF YEARS, FROM 1 TO 24 INClusive, at 5 and 6 per cent. |