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To find the interest due on notes, &c. when partial payments
have been made.
T 90. In Massachusetts the law provides, that payments shall be applied to keep down the interest, and that neither interest nor payment shall ever draw interest. Hence, if the payment at any time exceed the interest computed to the same time, that excess is taken from the principal; but if the payment be less than the interest, the principal remains unaltered. Wherefore, we have this RULE — -Compute the interest to the first time when a payment was made, which, either alone, or together with the preceding payments, if any, exceeds the interest then due; add that interest to the principal, and from the sum subtract the payment, or the sum of the payments, made within the time for which the interest was computed, and the remainder will be a new principal, with which proceed as with the first.
1. For value received, I promise to pay JAMES CONANT, or order, one hundred sixteen dollars sixty-six cents and six mills, with interest. May 1, 1822. $116,666.
On this note were the following endorsements :
Dec. 25, 1822, received $16'666
July 10, 1823,
What was due August 8, 1827 ?
Note. In finding the times for computing the interest, consult ¶ 40.
The first principal on interest from May 1, 1822, $116'666 Interest to Dec. 25, 1822, time of the first payment, (7 months 24 days,)
Amount, $121'215 16'666
Payment, Dec. 25, exceeding interest then due,
Remainder for a new principal,
15'490 Amount carried forward, $120'039
Amount brought forward, $120‘089 Payment, July 10, 1823, less than interest
Payment, Sept. 1, 1824, less than interest
Remainder for a new principal, (June 14, 1825,) Interest from June 14, 1825, to April 15, 1826, (10 months 1 day,)
Payment, April 15, 1825, exceeding interest then
Amount, $ 84'055
Remainder for a new principal, (April 15, 1826,) $22'055 Interest due Aug. 3, 1827, from April 15, 1826, (15 months 18 days,)
Balance due Aug 3, 1827,
2. For value received, I promise to pay JAMES LOWELL, or order, eight hundred sixty-seven dollars and thirty-three cents with interest. Jan. 6. 1820.
On this note were the following endorsements, viz.
April 16, 1823, received $136'44.
What remained due July 11, 1827 ? Ane. $215'103.
91. A promises. to pay B $256 in 3 years, with interest annually; but at the end of 1 year, not finding it convenient to pay the interest, he consents to pay interest on the interest from that time, the same as on the principal.
Note. Simple interest is that which is allowed for the principal only; compound interest is that which is allowed
for both principal and interest, when the latter is not paid at the time it becomes due.
Compound interest is calculated by adding the interest to the principal at the end of each year, and making the amount the principal for the next succeeding year.
1. What is the compound interest of $256 for 3 years, at 6 per cent.?
$256 given sum, or first principal.
15:36 interest, 256'00 principal, S
to be added together.
271'36 amount, or principal for 2d year.
16'2816 compound interest, 2d year, added t271'36 principal, do. gether.
287'6416 amount, or principal for 3d year.
17'25846 compound interest, 3d year,gether
first principal subtracted.
compound interest for 3 years.
2. At 6 per cent., what will be the compound interest, and what the amount, of $1 for 2 years? what the amount for 3 years?
for 4 years? for 7 years?
for 5 years? for 8 years?
Ans. to the last, $1'593+.
It is plain that the amount of $2, for any given time, will be 2 times as much as the amount of $1; the amount of
3 will be 3 times as much, &c.
Hence, we may form the amounts of $1, for several years, into a table of multipliers for finding the amount of any sun for the same time.
Showing the amount of $1, or 1£., &c. for any number of years, not exceeding 24, at the rates of 5 and 6 per cent. compound interest.
Years. 5 per cent.
1'15762+1'19101 + 1'21550+1626247+ || 16 1627628+1'33822 + 17 1'340091'41851 + 18 1'40710+ 1'50363+ || 19 81'47745+ 1′59384+|| 20 9 1'55132 1'68947+ 21 10 1'62889 + 1'79084 + 22 11 1710331'89829 +23
6 per cent.
2'653293'20713+ 2785963'39956 2'92526+3'60353 + 3'07152 + | 3'81974 +
12 179585201219+24 3622509 + ¦ 4′04893 +
Note 1. Four decimals in the above numbers will be sufficiently accurate for most operations.
Note 2. When there are months and days, you may first find the amount for the years, and on that amount cast the interest for the months and days; this, added to the amount, will give the answer.
3. What is the amount of $600'50 for 20 years, at 5 per cent. compound interest? at 6 per cent.?
$1 at 5 per cent., by the table, is $2'65329; therefore, 265329 600'50 $1593'30+ Ans. at 5 per cent.; and 3'20713 X 600'50 = $1925'881+ Ans. at 6 per cent.
4. What is the amount of $40'20 at 6 per cent. compound interest, for 4 years? for 10 years? for 18 years? for 12 years? for 3 years and 4 months? 18 days?
for 24 years, 6 months, and
Ans. to las, $168'137. Note. Any sum at compound interest will double itself in 11 years, 10 months, and 22 days.
From what has now been advanced we deduce the following general
I. To find the interest when the time is 1 year, or, to find the rate per cent. on any sum of money, without respect to time, as
the premium for insurance, commission, &c.,-Multiply the principal, or given sum, by the rate per cent., written as a decimal fraction; the product, remembering to point off as many places for decimals as there are decimals in both the factors, will be the interest, &c. required.
II. When there are months and days in the given time, to find the interest on any sum of money at 6 per cent.,-Multiply the principal by the interest on $1 for the given time, found by inspection, and the product, as before, will be the interest required.
III. To find the interest on $1 at 6 per cent., for any given time, by inspection,-It is only to consider, that the cents will be equal to half the greatest even number of the months; and the mills will be 5 for the odd month, (if there be one,) and 1 for every 6 days.
IV. If the sum given be in pounds, shillings, pence and farthings,-Reduce the shillings, &c. to the decimal of a pound, by inspection, (T 76;) then proceed in all respects as in federal money. Having found the interest, the decimal part, by reversing the operation, may be reduced back to shillings, pence and farthings.
V. If the interest required be at any other rate than 6 per cent., (if there be months, or months and days, in the given time,) -First find the interest at 6 per cent.; then divide the interest so found by such part or parts, as the interest, at the rate required, exceeds or falls short of the interest at 6 per cent., and the quotient, or quotients, added to or subtracted from the interest at 6 per cent., as the case may require, will give the interest at the rate required.
Note. The interest on any number of dollars, for 6 days, at 6 per cent., is readily found by cutting off the unit or right hand figure; those at the left hand will show the interest in cents for 6 days.
EXAMPLES FOR PRACTICE.
1. What is the interest of $1600 for 1 year and 3 months? Ans. $120.
2. What is the interest of $5'811, for 1 year 11 months? Ans. $'668. month 19 days, Ans. $'009.
3. What is the interest of $2'29, for 1 at 3 per cent.?
4. What is the interest of $18, for 2 years 14 days, at 7 per cent.?