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The interest accurately for 1 day is, of course, the interest for 1 year. For 2 days, then, it is

3 days,; for 47 days, and so on.

of ; for

7 of the interest for 1 year,

47 365

Hence, to find the interest accurately for any number of days: FIND WHAT FRACTIONAL PARt of 365 the dAYS ARE, AND MULTIPLY THE INTEREST FOR 1 YEAR BY IT.

NOTE. Reduce the Fraction to its lowest terms, before multiplying. This method is so tedious that it is little used. The common mode is usually as accu

rate as is necessary.

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1. Find the interest on $200.00 for 73 days. =200.06=$12.00. $12.00 ×4=$2.40 Ans. Ans. By com. meth. $2.433}.

365

2. Find the interest on 600.00 for 55 days.

NOTE.}•

365

Á. $5.4244.

Ans. By com. meth. $5.50.

3. A note lay at interest from the 15th Jan. 1824, to the 19th April, 1827. It was given for $1,025.67. What was its interest at the time of settlement?

A. $200.126623.

Ans. By com. meth. $200.68943.

NOTE. The error of the common mode was here greater on account of the intervention of February.

4. Find by both modes, the amount of $943.61 for 5 years, 47 d. 5. Find the interest, by both modes, on $1,864.00 for 6 yrs. 125 d. 6. What was the amount of a note for $7,684.331, which had been lying on interest from Dec. 19th, 1811, to May 27th, 1825, by both modes?

§ LXXX. We mentioned (§LXXVII.) that it was easier to find interest at 6 pr. ct. than at any other rate. The pupil is now prepared to understand the reason of this; which is, that 6 is just half the number of months in the year. To find interest at any other rate, for months and days, let it be remembered, that

1 pr. ct. is of 6 pr. ct.

2 pr. ct. is 21 of 6. 3 pr. ct. is 31 of 6. 4 pr. ct. is 4 of 6.

5 pr. ct. is of 6.

66 6.

7 pr. ct. is 7-11 times 6.
8 ct. is 8=4=11
9 pr. ct. is 2-3=11

pr.

10 pr. ct. is

66 6.

66

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==13 6. 66 6.

11 pr. ct. is=1{

In this manner, may be found what part any rate is of 6 pr. ct. Hence, we have a universal rule for interest at any rate pr. ct.

1. FIND THE RATE FOR THE WHOLE TIME AT 6 PR. CT., TAKE SUCH PART OF THIS AS THE GIVEN RATE IS OF 6 PR. CT., AND MULTIPLY THE GIVEN PRINCIPAL BY IT. Or the following:

II. FIND THE INTEREST AT 6 pr. ct., anD TAKE SUCH PART OF IT AS THE GIVEN RATE IS OF 6 pr. ct.

We leave abbreviations to the ingenuity of the pupil.

EXAMPLES FOR PRACTICE.

1. What is the interest on $29.25 for 4 months, at 3 pr. ct. ? NOTE. 3 pr. ct. is of 6 pr. ct. Therefore, find the interest at 6 pr. ct. and multiply it by . A. $0.2921.

2. Find the interest for 3 months on $72.05, at 2 pr. ct.

A. $0.3601.

3. Find the interest on $600.00, at 4 pr. ct., for 4 months and 18 days? A. $9.20. 4. Find the interest on $9,000.00, at 5 pr. ct., for 8 mo. 4 d. ? A. $305.00.

5. Find the interest on $800.00 for 6 mo. 21 d., at 7 pr. ct. ?

A. $31.2663.

6. Find the interest on $1,000.00, at 8 pr. ct., for 3 yrs. 6 mo. 9 d.

A. $281.00.

7. Find the interest on $1,200.00, for 1 yr. 3 mo. 15 d. at 9 pr. ct. A. $139.50.

8. Find the interest on $625.00, at 10 pr. ct., for 3 yrs. 2 mo. 7 d. A. $199.131.

9. Find the interest on $930.00, for 6 yrs. 3 mo. 11 d., at 11 pr. ct. A. $642.500.

10. Find the interest on $780.00, for 9 yrs. 7 mo. 16 d. at 12 pr. ct. 11. Find the interest on $1,825.25, for 6 yrs. 8 mo. 13 d., at 13 pr. ct.

12. Find the interest on $3,976.18, for 2 yrs. 4 mo. 8 d., at 8 pr.

ct. 13. Find the interest on $1,964.43, for 9 yrs. 5 mo. 13 d. at 7 pr. ct. 14. Find the interest on $2,675.33, for 7 yrs. 7 mo. 7 d. at 5 pr. ct. 15. Find the interest on $3,865.49, for 12 yrs. 3 mo. 5 d. at 7 pr. ct. Besides the modes given above, there is another, which is often convenient. We have seen how to compute the interest for even years at any rate. (§ LXXVI.) A rate for the whole time is found, by multiplying the rate pr. an. by the number of years. Now if we reduce months and days to decimals of a year, we can treat them in the same way. Thus,

16. Find the interest on $100.00, for 2 yrs. 6 mo., at 7 pr. et. A. $17.50. 2 yrs. 6 mo. 2 yrs.=2.5 yrs. Then, 2.5.07.175 rate for whole time.

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17. Find the interest on $700.00, for 3 yrs. 2 mo. 12 d., at 5 pr. ct. A. $112. The time is 3.2 yrs. Therefore, the rate for whole time is 3.2× .05.16.

Hence, the rule is

III. MULTIPLY THE RATE PER CENT. PER ANNUM BY THE TIME IN YEARS AND DECIMALS OF A YEAR; THE PRODUCT WILL BE THE RATE FOR THE WHOLE TIME, WITH WHICH PROCEED AS USUAL.

Or,

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

et.

LXXXI.

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 INTE

REST are given, to find the PRINCIPAL.

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

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

The

If the rate be 6 pr. ct. pr. an. it is 12 pr. ct. for two years. 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 decimals. § LXXX, Rule 111. 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, and TIME are given, to find the RATE.

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 I 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 mc. 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: 8488=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.

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 5 years, when the interest was $45.00. Required the rate pr. ct. pr. an.

45—15.15 rate for 5 years.

300 100

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. Required the rate pr. ct. pr. an.

761

2893.13 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 FIGURE

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 .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,

÷=.083 of a year.

7 mo.÷=.583 of a year. `1 mo.÷

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.

1

II, FIND THE RATE FOR THE WHOLE TIME, and divide iT BY THE RATE GIVEN.

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