6. What is the difference between the discount of $100.00 for 1 year, and the interest of the same sum, for the same time? Ans. $0.33933.

7. A note was given without interest for 13 months, but at the end of 5 months the debtor paid the note at a discount of 6 pr. ct. How much did he pay, the note being for $847.00. A. $822.3313· 8. I pay a debt of $394.00 1 year and 8 mo. before it is due. What ought I to pay? A. $358.1811T 9. If a debt of $137.00 be paid 2 yrs. 6 mo. before it is due what ought to be paid ? A. $114.166.

10. What is the difference between the discount and the interest of $500.00 for 4 yrs.? A. 23.225.

11. What is the discount on $100.00 for 1 yr. 6 mo. at 5 pr. ct, ? Rate at pr. ct.-.09% of .09 (§ Lxxx.)=.075 rate at 5 pr. ct.

A. $93.02311. 12. What is the present worth of $227.26 for 4 yrs. 6 mo. at 7 A. $172.82173. pr. ct.

13. What is the present worth of $1,000.00. for 9 yrs., 9 mo., and 18 d., at 9 pr. ct.?

14. What is the present worth of $7,963.88 for 13 yrs. 7 mo. 6 d. at 4 pr. ct.?

15. What is the present worth of $1,759.40, for 10 yrs. at 11 pr. ct. ?

The principles of discount, as we have exhibited them, have their foundation in common sense. It is perfectly manifest, that the discount on a sum of money ought never to be equal to the interest on the same sum for the same time. For the discount is the interest on the present worth, which is, of course, less than the sum itself. Yet, it is customary, in banks, to take the full interest on a note in discounting. This is evidently usury. For example, if I present a note for $100.00, due 6 months hence, for discount, the bank allows me only $97.00 taking $3.00, the full interest on the note for 6 months, as discount. Now if I put $97.00 at interest for 6 months, it only amounts to $99.91, i. e. 9 cts. less than the note. If the time were a year, I should receive only $94.00, 6 dollars, being reserved for discount. $94.00 in a year, amounts to $99.64, being 36 cents less than the note. If it were 2 years, the error would be $1.44, and so on, increasing as the square of the time. [See § XCVIII. for the meaning of the word square.] Banks do not, however usually discount for so long a period. For $1,000.00, this last error would become $14.40. For $100,000.00, it would be $1,440.00.

=

When discounts are made for 6 months in this manner, the rate per cent. is actually, 6 more than 6,2 pr. ct. When for 1 year, the rate is 61 pr. ct.=more than 6, and almost 63 pr. ct. When for 2 years, the rate is 6 pr. ct.=more than 62, and almost 61 pr. ct.

Yet, notwithstanding these facts, the supreme courts have decided, that this mode of calculating discount is not usury. Banks are, therefore, authorized by judicial authority to discount at a higher rate than that established by law. We know not on what principles this decision has been made.

$ LXXXIII. It will often be found convenient, even when the rate is 6 pr. ct. to find the interest on a sum for 12 pr. ct. pr. an., and to take such part of the interest, thus found, as the given rate may be of 12 pr. ct. The reason of this is, that there are 12 months in the year, and therefore the rate will be 1 pr. ct. a month. In finding the rate for odd numbers of months, at 6 pr. ct., we get two figures; one in the hundredthis', and one in the thousandths' place. Thus for 7 months, at 6 pr. ct., the rate is .035; for 9 months, .045, &c. But at 12 pr. ct. these same rates will be .07 and .09. Thus, in the latter case, we have but one figure in the multiplier.

1. Find the interest on $962.54 for 9 mo. at 6 pr. ct.

.09-rate at 12 pr. ct. .09×962.54-86.6286. 86.6286+2= 43.3143 Ans.

2. Find the interest on $862.00 for 7 mo. at 4 pr. ct.

.07=rate at 12 pr. ct. .07×862=60.34 60.34÷3=20.113 Ans. 3. Find the interest on $365,25 at 3 pr. ct. for 5 months.

In many cases where days occur, we obtain, by this method, a similar advantage.

4. Find the interest on $324.00 for 7 mo. 15 d.

15 days are half a month. At 12 pr. ct., then, the rate is .075. It would be .0375 at 6 pr. ct., which is not so convenient, .075×324 24.30 24.30÷2=$12.15 Ans.

5. Find the interest on $800.00, at 8 pr. ct. for 5 mo. 21 d. The rate at 12 pr. ct. is .057 .057×800=45.60×3=$30.40 Ans. 6. Find the interest on $9,823, for 11 mo. 27 d. at 9 pr. ct. 7. Find the interest on $8,763.25 at 2 pr. ct. for 3 mo. 9 d. 8. Find the interest on $11,864 for 13 mo. 3 d.

LXXXIV. In calculating interest on Sterling Money, or upon the U. S. currencies, it is best to reduce the lower denominations to decimals of a pound. For this purpose, it is sufficiently correct, and much the most convenient mode to employ the contraction in § LX. After the calculation is made, reduce the resulting decimal to shillings, &c. again, by the contraction § LXIV. 1. Find the interest on £13; 3; 6 for 1 year. A. 15s. 9d. 2 qrs. 2. Find the interest on 13£ 15s, 3d. 2 qrs. for 1 yr. 6 mo.

A. £1; 4; 9; 1. 2 mo. A £23; 7; 7. 6 mo. A. £36; 13.

3. Find the interest on £75; 8; 4 for 5 yrs. 4. Find the interest on £174; 10; 6 for 3 yrs. 5. Find the interest on £325; 12; 3 for 5 yrs. A. £97; 13; 8. 6. Find the interest on £150; 16; 8 for 4 yrs. 7 mo. A. £41; 9; 7. 7. Find the amount of £3,000, for 12 yrs. 10 mo. A. £5,310. 8. Find the amount of £279; 13; 8 for 3 yrs. 6 mo. at 5 per ct. A. £331; 1; 6.

9. Find the interest on £137; 17; 2 from Jan. 11th, 1822, to Aug. 15th, 1822. A. £4; 18; 4.

10. Find the interest on £43; 16 for 9 mo. 13 d. at 8 pr. ct.

A. £2; 15; 1. 11. Find the interest on £87; 15; 4 for 2 yrs. 11 mo. 3 d, at 71⁄2 pr. ct. A. £19; 5; 2.

12. Find the interest of £137; 11 for 11 d. at 9 per ct.

A. 7s. 6d. 3 qrs.

13. Find the interest on £16;7; 8; for 2 mo. at 12 per ct. A. 6s. 6d. 3 qrs.

14. Find the interest on 15s. for 3 mo. at 8 per ct.

A. 3d. 2 qrs.

15. Find the interest on £193; 2; 6; 1, for 9 years..
16. Find the interest on £967; 18; 3, for 16 yrs. 4 mo. 11 d.
17. Find the interest on £1,000, for 5 yrs. 3 mo. 4 d.

18. Find the interest on £2,030; 17; 6; 1, for 6 yrs. 4 mo. 3 d. 19. Find the interest on £1,894; 0; 0; 1, for 19 d.

20. Find the interest on £6,872; 0; 3, for 17 d.

21. Find the interest on £15; 15; 5, for 8 hours.

22. Find the interest on £17; 13; 9; 2, for 3 yrs. 2 d. at 11 per ct. 23. Find the interest on £225; 17; 9: 1 for 15 d. at 7 per ct. 24. Find the interest on £ 8,857; 16; 3; 2 for 27 d. 15 hours.

$ LXXXV. In the calculation of profit and loss, where credit is given, it is plain, that we must discount for the time that intervenes between the bargain and payment, in order to form a correct conclusion. To be perfectly accurate, we should likewise calculate the interest on the cost, for the time the goods have lain useless in the hands of the seller.

1. Sold goods to the amount of $725.00 at 6 months credit. By the bargain I gain 25 dolls. What is my true profit?

Ans. $3.883

2. I make sales to the amount of $654.37. The goods cost me $600.00, and have been lying a year on my shelves. I give credit for 8 months. Do I gain or lose, and how much?

I apparently gain $54.37. But I must deduct $25.168 for discount, and likewise $36.00 for interest on the cost. These deductions amount to $61.168, so that in fact I lose $6.793, instead of gaining as I supposed.

As money will not always command the legal interest, however, the dednotions ought to be made at the rate pr. ct. which can be obtained for it.

3. I have merchandize on hand for 6 months, and at last am able to sell it at an advance of 25 pr. ct. on the cost, giving 4 months credit. The amount of my sales is $1,000.00. What is my gain supposing that money will command 5 pr. ct. interest?

Ans. $163.60631,

4. If I sell goods for 15 pr. ct. advance on the cost, ready cash, to to the amount of $1,680.00, when money will command only 4 pr. ct., what is my gain, supposing I have had the goods on hand 1 yr. 6 mo. ?

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?

367122072°

Ans. Whole gain $566.049225 1 8 2 17 7. -On each art. $51.93211-$487.660355.-$26.456187.--Rates .10. -,143 53. ———. 1735.—Average rate .14,196445181689

2954672385818

$ 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 usual 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-$118.3431. 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 13 pr. ct, of which, the invoice is $654.59. The second of the above modes is recom. mended. 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, 654.59×131=$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. 122 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.