IV. MULTIPLY THE PRINCIPAL SUCCESSIVELY BY THE TIME AND RATE PER CENT. PER ANNUM. THE LAST PRODUCT IS THE INTEREST REQUIRED. Some rules for finding the decimal of time, may be found in § LXXXI.

ct.

18. Find the interest on $1,875.31, for 7 yrs. 6 mo. 4 d., at 8 pr.

19. A note for $2,633.75, was at interest at 7 pr. ct. from the 28th of September, 1809, to the 15th of March, 1829, What was

then due ?

VARIOUS CALCULATIONS IN INTEREST.

LXXXI. ART. I. When the TIME, RATE, and INTEREST are given, to find the PRINCIPAL.

When no particular rate is mentioned, 6 pr. ct. must be understood, through out these exercises.

1. What principal will gain 60 dolls. in 2 years?

If the rate be pr. ct. pr. an. it is 12 pr. ct. for two years. The question then is, 60 is 12 pr. ct. on what principal? Ans. $500.00. 2. What principal will acquire $459.45 in 25 years 6 mo. 9 d. ? The rate for the whole time is 1.5315. Ans. $300.00. 3. What principal will acquire $934.20 in 17 yrs. 3 mo. 18 d.? Ans. $900.00. 4. What principal will give an interest of $72.25 in 3 yrs. 5 mo. 25 d.?

5. What principal will gain $965:30 in 18 yrs. 9 mo. 3 d. ? 6. What principal will acquire $1,000.00 in 16 yrs. 8 mo. ? Hence, the rule is,

FIND THE DECIMAL RATE FOR THE WHOLE TIME, AND DIVIDE THE

GIVEN INTEREST BY IT.

NOTE-For any other than 6 pr. ct. find the rate for the whole time by deci mals. § LXXX, Rule III. Or find the rate at 6 pr. ct., and take such part of this rate as the given rate may be of 6 pr. ct.

7. A note lay 12 yrs. 3 mo. 20 d. and at the end of the time there was an interest on it of $350.27. What was the principal of the note?

8. A note was given April 29th, 1820, and settled Nov. 13th, 1826. At that time the interest was $540.00. What was the principal of the note?

ART. II. When the PRINCIPAL, INTEREST, are given, to find the RATE.

and TIME

1 A man has $2,000.00 at interest, and at the end of 3 yrs. the interest is $300. What pr. ct. pr. an. does he receive?

If the rate had been 1 pr. ct. pr. an., the interest on $2,000.00 would in three years have been 2,000.03=$60.00. As often, then, as $60.00 is contained in the actual interest $300.00, so many times 1 pr. ct. will there be in the rate required.

$300÷60-5 pr. ct. the rate required,

2. $720.00 lay at interest 7 yrs. 2 mo. and 12 d. then was $311.04. What was the rate?

At 1 pr. ct. the principal would have gained $51.84. 84-6 pr. ct. Ans.

The interest

311.04÷51.

3 $300.00 in 2 years gained $42.00. What was the rate of interest allowed?

4. $960.00 gained $180 in 2 yrs. 6 mo. 23 d.

interest was allowed?

Hence, the rule seems to be,

Ans. 7 pr. ct.

What rate of

DIVIDE THE GIVEN INTEREST BY THE INTEREST ON THE SAME PRINCIPAL, FOR THE SAME TIME, at 1 per cent.

4. $750.00 in 12 yrs. 3 mo. gained $551.25.

pr. ct. pr. an. ?

What was the rate
Ans. 6 pr. ct.

5. A note given for $2,365.00, on the 19th of June 1826, was paid on the 14th of July 1829. The interest then due was $435.554. What rate was allowed?

6. $20,000.00 lay 73 years at interest. $58,400.00. What was the rate allowed?

Ans. 6 pr. ct.

The interest then was

A different mode from the last may be proposed, which in many instances will be better. Since $58,400.00 is the interest on $20,000.00 for the whole time, we may find the rate for the whole time, thus: 888-2.92-rate pr. ct. for 73 yrs. If this be divided by the number of years, we shall have the rate for 1 year.

58400

2.92 73-4 pr. ct. the rate required. This saves us the trouble of calculating the interest at 1 pr. ct. for a great number of years. It is likewise convenient for shorter periods.

7. $300.00 was at interest years, when the interest was $45.00. Required the rate pr. ct. pr. an.

15 15 rate for 5 years. This is 3 pr. ct. for 1 yr. Ans. 8. The interest on $761.00 was $98.93 for 2 yrs. 7 mo. 6 d. quired the rate pr. ct. pr. an.

76100

Re

893.13 rate for whole time. But the time is not even years. Therefore we must reduce the months and days to a decimal before dividing. 2 yrs. 7 mo. 6 d.=2.6 (§ Lx.)

.13 2.6-.05=5 pr. ct. Ans. 9. A note for $930.25 was given on the 11th March 1823, and on the 29th of May, 1824, its interest was $111.63. What was the rate? Ans. 10 pr. ct.

Hence, we have the rule,

II. FIND THE RATE FOR THE WHOLE TIME AND DIVIDE IT BY THE TIME IN YEARS AND DECIMALS.

In determining which rule to use, the judgment of the pupil must be exercised. Some questions are solved by the last, much more expeditiously, than by the other; and, on the other hand the first is often the most convenient. Some concise modes of finding the decimal of a year may be given. In calculating interest we allow 30 days to the month, and 12 months to the year; making 360 days in a year. 1 tenth or .1 of a year, then, is 36 days. To reduce months and days to the decimal of a year, therefore,

BRING THE WHOLE TO DAYS AND DIVIDE BY 36, ANNEXING CYPHERS IP NECESSARY. IF THE DAYS ARE 36 OR MORE, THE FIRST QUOTIENT FIGURR

WILL BE TENTHS; IF NOT, IT WILL BE HUNDREDTHS, OR THOUSANDTHS, ACCORDING AS ONE OR TWO CYPHERS ARE ANNEXED, TO OBTAIN IT.

Thus, 3 mo. 24 d.=114 d. 114÷36=316 of a year.

9 d.÷36=.025 of a year. 2 d.÷36=.005 of a year.

Another mode may be given for months. 6 months is 1.5 of a year. Then, .1 of a year is six fifths of a month.

Hence, to reduce months to the decimal of a year,.

DIVIDE THE NUMBER OF MONTHS BY SIX FIFTHS ANNEXING CYPHERS IF NECESSARY. THE FIRST QUOTIENT FIGURE, IF FOUND WITHOUT ANNEXING A CYPHER, IS TENTHS; IF A CYPHER IS ANNEXED TO OBTAIN IT, HUNDREDTHS.

Days, with or without months, may be made Fractions of a month, vulgar, or decimal, and treated in the same manner. Or, the decimal for the months may be found by this rule, and that for the odd days, by the last, and the two results added. Thus,

7 mo.÷=.583 of a year. 1 mo.÷=.083 of a year.

9 mo. 6 d. 9.2 mo.+=.76 of a year. 9 d.=.3 mo.÷=.025 of a year.

10. A note was given on the 7th Jan. 1823, for $375.50 and paid on the 16th Oct. 1824, at which time the interest was $53.321. What rate was allowed ? Ans. 8 pr. ct.

11. A note was given on the 4th March 1818, for $2,000.00, and paid on the 16th May 1829, when the interest was $1,120.00. What rate pr. ct. pr. an. was allowed?

12. A note was given on the 17th June 1827, for $2,000.00, and paid on the 17th Dec. 1829, when the interest was $200.00. What rate was allowed ?

ART. III. When the PRINCIPAL, RATE, and INTEREST are given to find the TIME.

1. $300.00 gains $54.00. How long In one year $300.00 will gain $18.00. in as many years, as there are 18s in 54. 2. $1,000.00 gains $300.00 at 5 pr. ct.

interest?

has it been at interest?
It will gain $54.00 then
54-18-3 years. Ans.
How long has it been at

In one year $1000.00 will gain $50.00.
3. How long will it take $500.00 to gain
Hence, the rule is,

300+50-6 yrs. Ans. $350.00, at 7 pr. ct.?

I. DIVIDE THE GIVEN INTEREST, BY THE INTEREST ON THE SAME PRINCIPAL, For 1 year.

NOTE. When a Fraction is obtained by this division, it must be reduced to months and days.

4. How long will $2,000.00 be in gaining $414.00 ?

A. 3 yrs. 5 mo. 12 d. 5. How long will $200.00 be in gaining $36.00? A different mode from the last may be employed. Let the rate be found for the whole time, as in ART. II. Thus, 3.18. Now as .18 is the rate for the whole time, if I divide by the rate for year, I shall find the number of years. .18.06 3. Ans. 3 yrs. Hence, we have another rule.

I

II. FIND THE RATE FOR THE WHOLE TIME, AND DIVIDE IT BY THE BATE GIVEN.

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 DIVIDE 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 $2,515.93. 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

The ques

interest, to $318.00, at the end of the year. tion 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. 1n 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 BÝ IT. The DISCOUNT will be found by subtracting the present worth from the whole debt.

6

00

069

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 C 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. 10, 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.

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.