Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

Fractions, as before, must be reduced to months and days.

6. How long will $600.00 be in gaining $30.00? Ans. 10 mo. 7. A note was given on the 11th of Jan. 1819, for $754.50, and when it was paid the interest was $58.866. On what day was it paid?

IV. When the AMOUNT, RATE, and TIME are given, to find the PRINCIPAL.

1. A sum of money has been on interest 2 years, and it amounts to $112.00. What is the sum?

6 pr. ct. for 1 year is 12 pr. ct. for 2 years. The question then is, what principal will amount to $112.00 at 12 pr. ct.

In $112 is contained once the principal, and .12 of it besides. To the rate .12 then, if I add a unit, making 1.12, this number will express how often the principal is contained in the amount. Of course I must divide the amount by it. $112÷1.12=$100. Ans. 2. A sum of money in 3 yrs. 9 mo. 12 d. amounts to $1,227.00. What is the sum ? Ans. $1,000.00. 3. A note in 2 yrs. amounted to $275.00 at 5 pr. ct. What was the principal of the note?

Hence, the rule,

FIND THE RATE FOR THE WHOLE TIME, ADD TO IT A UNIT, AND DI

VIDE THE GIVEN AMOUNT BY THE SUM.

NOTE. For any other than 6 pr. ct. the rate for the whole time may be found by decimals. § LXXX. Rule III. Or the rate at 6 pr. ct. may be found, and such part of it taken, as the given rate is of 6 pr. ct. The interest may be found by subtracting the principal from the amount. 4. An amount for 3 yrs. 6 mo. 15 d. was $2,515.931. What was the principal ? Ans. 2,075.00

5. A note was given on the 10th Sept. 1825, and paid on the 17th May 1829, at which time $1,375.988833 was due. For what sum was the note given?

6. A note was given for 6 yrs. 8 mo. 27 d. with interest. When paid, it amounted to $30,628.33. What was the principal ?

§ LXXXII. DISCOUNT. 1. I have a note of $318.00 due me one year from the present date without interest; but my debtor is willing to pay me now, if I will allow discount at 6 pr. ct. 1 agree to this, and he pays me. What sum do I receive ?

The pupil should consider that I have $318.00 due me at the end of the year, and not before. Of course it makes no difference to me, whether my note is $318.00 exactly, without interest, or whether it is for some other sum, which will amount to $318.00 at the end of the year. For, in either case, I receive, actually, the same amount, at the same time. Hence, the sum which my debtor pays me ought to be such as would amount, at

interest, to $318.00, at the end of the year. The question then is, what principal will amount to $318.00 in one year. It therefore belongs to § LXXXI. ART. IV. Ans. $300.00. This $300.00 which my debtor should fairly pay me, at the present time, instead of the whole debt at a future day, is called the PRESENT WORTH of the debt. Then, when money is due, at a future day, without interest,

THE SUM WHICH, AT INTEREST, FOR THE GIVEN RATE AND TIME, WOULD AMOUNT TO THE SUM THEN DUE, IS CALLED THE PRESENT WORTH OF THAT SUM.

The DISCOUNT on the debt, or the deduction made, on account of present payment, then, is the interest on the present worth, and not on the whole debt, for the given time. Therefore,

The PRESENT WORTH may be considered as a PRINCIPAL, the DISCOUNT, the INTEREST, for the given time, on that principal, and the DEBT itself, the AMOUNT.

2. In 2 yrs. 6 mo. I have $1,000.00 due me. worth of the debt? A. $850.00.

What is the present

3. What is the present worth of $725.25 for 3 yrs. 5 mo. 8 d.? Hence, the rule, to find the present worth,

FIND THE RATE FOR THE WHOLE TIME, ADD TO IT A UNIT, AND DIVIDE THE GIVEN SUM BY IT. The DISCOUNT will be found by subtracting the present worth from the whole debt.

NOTE. For any other rate than 6 pr. ct. it is best to use decimals § LXXX. Rule III. Or we may find the rate for the whole time at 6 pr. ct. and take such part of this rate, as the given rate may be of 6 pr. ct. by the rules in the same §. NOTE. A decimal multiplier, for calculating discount, may be found thus. Any sum is 188 of its amount for 1 year, at 6 pr. ct. And its interest, for the same time, is of the same amount. Hence, the present worth of a sum, for 1 year, at 6 pr. ct., is 188, and the discount, for the same time, T of the sum itself. 188, then, reduced to a decimal, will give a multiplier for finding the present worth; and reduced to a decimal will give a multiplier for finding discount. For any other rate and time, the process is similar. When the decimal multiplier is obtained, discount may be calculated in the same manner as interest.

4. What is the present worth of $600.00 due 2 yrs. 6 mo. hence at 5 pr. ct. discount?

2 yrs. 6 mo. 2. 5. yrs. 2.5X.05.125 rate for whole time. Or 15-rate for whole time at 6 pr. ct. (§ LXXVIII.) 5 pr. ct. is of 6 pr. ct. .15x=.125 and 600÷1.125=533.333 Ans.

3

5. What is the discount of $100.00 for 1 year? A. 5.66 .06=rate for 1 yr. Then 1+.06-1.06 and 100÷1.06=93.33.

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. $822.33. 8. I pay a debt of $394.00 1 year and 8 mo. before it is due. What ought I to pay? A. $358.181. 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. 23.2253.

11. What is the discount on $100.00 for 1 yr. 6 mo. at 5 pr. ct.? Rate at 6 pr. ct.=.091⁄2 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.

77

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, 61=more than 6 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 62 pr. ct.=more than 63, and almost 6 pr. ct.

22

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 hundredths', 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. 43.3143 Ans.

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

86.6286÷2

.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.

A £23; 7; 7.

3. Find the interest on £75; 8; 4 for 5 yrs. 2 mo. 4. Find the interest on £174; 10; 6 for 3 yrs. 6 mo. A. £36; 13. 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.

11. Find the interest on £87; 15; 4 for 2 yrs. pr. ct. A. £19; 5; 2.

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

A. £2; 15; 1.

11 mo. 3 d, at 7

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.

A. 3d. 2 qrs.

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

16

Ans. $3.883 51 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 discount, and likewise $36.00 for interest on the cost.

for

These de

ductions 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 deductions 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 eredit. The amount of my sales is $1,000.00. What is my gain supposing that money will cominand 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. ?

« ΠροηγούμενηΣυνέχεια »