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206. Problem V. To find the PRINCIPAL, the interest, time, and rate per cent. being given.
Ex. 1. What principal at 6 per cent. will gain $36 in 2 years? Ans. $300.
We find the interest of $1 for 2 years, by which we divide the given interest.
.0 6 int. of $1 for ly.
.12 ) $3 6.0 0 ( $3 0 0 principal.
Since it requires 2 years for a principal of $1 to gain 12 cents, it will require a principal
of as many dollars to gain $36 as $0.12 is contained times in $ 36, or $ 300.
- Divide the given interest or amount by the interest or amount of $1 for the given rate and time, and the quotient will be the principal.
EXAMPLES FOR PRACTICE.
2. What principal will gain $24.225 in 4 years, 3 months, at 6 per cent.? Ans. $95.
3. What principal will gain $5.11 in 3 years, 6 months, at 8 per cent.? Ans. $18.25.
4. The interest on a certain note at 9 per and 8 months amounted to $42; what was of the note?
cent. in 1 year the full amount Ans. $280.
207. Compound Interest is interest on both principal and interest, when the latter is not paid on becoming due.
The law specifies that the borrower of money shall pay the lender a certain sum for the use of $100 for a year. Now, if he does not pay this sum at the end of the year, it is no more than just that he should pay interest for the use of it as long as he shall keep it in his possession. The computation of compound interest is based upon this principle.
206. What is Problem V.? Explain the operation. The rule for finding the principal, the interest, time, and rate per cent. being given? · What is compound interest? On what principle is it based?
208. To find the compound interest of any sum.
Ex. 1. What is the compound interest of $500 for 3 years, 7 months, and 12 days, at 6 per cent. ?
Interest of $1 for 1 year,
Interest for 1st year,
Amount for 1st year,
Interest for 2d year,
Amount for 2d year,
Interest for 3d year,
Amount for 3d year,
Interest of $1 for 7mo. 12 da.,
Interest for 7mo. 12da.,
Amount for 3y. 7mo. 12da.,
We first find the interest of the principal for 1 year, and add the interest to the principal for a new principal. We then find the interest of this principal for 1 year, and proceed as before; and so also with the third year. For the months and days we find the interest on the amount for the last year, and, adding it as before, we subtract the original principal from the last amount for the answer.
RULE. Find the interest of the given sum for one year, and add it to the principal; then find the amount of this amount for the next year; and so continue, until the time of settlement.
If there are months and days in the given time, find the amount for them on the amount for the last year.
Subtract the principal from the last amount, and the remainder is the compound interest.
208. Explain the operation in computing compound interest. The rule?
NOTE 1. If the interest is to be paid semi-annually, quarterly, monthly, or daily, it must be computed for the half-year, quarter-year, month, or day, and added to the principal, and then the interest computed on this, and on each succeeding amount thus obtained, up to the time of settlement.
NOTE 2. When partial payments have been made on notes at compound interest, the rule is like that adopted in Art. 199.
EXAMPLES FOR PRACTICE.
2. What is the compound interest of $761.75 for 4 years? Ans. $199.941. 3. What is the amount of $67.25 for 3 years, at compound interest? Ans. $80.095. 4. What is the amount of $78.69 for 5 years, at 7 per cent.? Ans. $110.364. 5. What is the amount of $128 for 3 years, 5 months, and 18 days, at compound interest? Ans. $156.717.
6. What is the compound interest of $76.18 for 2 years, 8 months, 9 days? Ans. $12.967. 209. Method of computing compound interest, by
means of a
TABLE SHOWING THE AMOUNT OF $1, OR £1, FOR ANy Number of Years, from 1 TO 20, AT 3, 4, 5, 6, AND 7 PER CENT., COMPOUND INTEREST.
3 per cent. 4 per cent.
5 per cent.
6 per cent.
7 per cent.
1.229873 1.315931 1.407100
1.652847 1.947900 2.292018
2.750032 15 2.952164 16 3.158815 17 18 19 20
209. If the interest is to be paid semi-annually, quarterly, &c., how is it computed? How, when partial payments have been made?
Ex. 1. What is the interest of $240 for 6 years, 4 months, Ans. $107.593.
and 6 days, at 6 per cent.?
We multiply the principal by the amount of $1 for 6 years in the table, and obtain the amount for 6 years. We then find the interest on this amount for the 4 months and 6 days, and add it to its principal, and from the sum subtract the principal for the answer. Hence,
Multiply the amount of $1 for the given rate and time, as found in the table, by the principal, and the product will be the amount. Subtract the principal from the amount, and the remainder will be the compound interest. If there are months and days in the time, cast the interest for the months and days as in the foregoing rule.
EXAMPLES FOR PRACTICE.
2. What is the interest of $884 for 7 years, at 4 per cent.? Ans. $279.283. 3. What is the interest of $721 for 9 years, at 5 per cent.? Ans. $397.507. 4. What is the amount of $960 for 12 years, 6 months, at 3 per cent.? Ans. $1389.26.
5. What is the amount of $25.50 for 20 years, 2 months, and 12 days, at 7 per cent.? Ans. $100.058.
6. What is the amount of $12 for 6 months, the interest to be added each month? Ans. $12.364+.
7. What is the amount of $100 for 6 days, the interest to be added daily? Ans. $100.10004.
210. Discount is an allowance or deduction for the payment of a debt before it is due.
The Present Worth of any sum is the principal, which, being put at interest, will amount to the given sum in the time for which the discount is made. Thus, $100 is the present worth of $106, due one year hence at 6 per cent.; for $100 at 6 per cent. will amount to $106 in this time; and $6 is the discount.
NOTE.- Business men, however, often deduct five per cent., or more, from the face of a bill duc in six months, or a percentage greater than the legal rate of interest.
211. The interest of any sum cannot properly be taken for the discount; for the interest for one year is the fractional part of the sum at interest, denoted by the rate per cent. for the numerator, and 100 for the denominator; and the discount for one year is the fractional part of the sum on which discount is to be made, denoted by the rate per cent. for the numerator, and 100 plus the rate per cent. for the denominator. Thus, if the rate per cent. of interest is 6, the interest for one year is 8 of the sum at interest; but if the rate per cent. of discount is 6, the discount for one year is T8 of the sum on which discount is made.
212. In discount, the rate per cent., time, and the sum on which the discount is made, are given to find the present worth.
These terms correspond precisely to Problem VI. in interest, in which the rate per cent., time, and amount are given to find the principal. (Art. 203.)
213. To find the present worth and the discount of any sum due at a future time.
Ex. 1. What is the present worth of $25.44, due one year hence, discounting at 6 per cent.? What is the discount?
Ans. $24 present worth; $1.44 discount.
210. What is discount? The present worth of any sum of money? How illustrated? - 211. Are interest and discount the same? Explain the difference Which is the greater, the interest or discount on any sum, for a given time?-212. What terms are given in discount, and what is required? To what do these correspond in interest?