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68. Discount is an allowance made for the payment of money before it is due.
The present worth of a debt, payable at some future time, without interest, is such a sum of money as will, in the given time, amount to the debt.
When the interest is at 6 per cent., the amount of $1, for 1 year, is $1.06 ; therefore the present worth of $1.06, due 1 year hence, is $1. We may also infer that the present worth of any sum for 1 year, will be as many dollars as $1.06 is contained in the given sum. Hence we have the fol. lowing
RULE. Find the amount of $1, for the given time, at the given rate per cent., then divide the sum by this amount, and it will give the present worth. Subtract the present worth from the amount, and it will give the discount.
Examples. 1. What is the present worth of $622.75, due 3 years and six months, at 5 per cent. ?
In this example we find the amount of $1, for 3 years and 6 months, at 5 per cent., to be $1.175 ; therefore dividing $622,75 by $1.175, we get $530, for the present worth. If we subtract the present worth from the sum, we get $92.75 for the discount.
2. What is the present worth of $4161.575, due 3 months hence, at 9 per cent. ?
Ans. $4070. 3. What is the present worth of $7.10272, due 4 years and 12 days hence, at 8 per cent. ?
4. Sold goods for $1500, to be paid one-half in 6 months, and the other half in 9 months; what is the present worth of the goods, interest being at 7 per cent.? ,
Ans. $1437.226. 5. What is the present worth of $50, payable at the end of 3 months, at 7 per cent. ?
. Ans. $49.14. 6. What is the discount on $100, due 6 months hence, at 6 per cent.?
Ans. $2.9:13. 7. What is the discount on $750, due 9 months hence, at 7 per cent.?
Ans. $37.411. 8. What is the present worth of $3471.20, due 3 years and 9 months hence, at 45 per cent.? . Ans. $2970.01.
9. What is the discount of $150, due 3 months and 18 days hence, at 6 per cent. ?
10. What is the discount of $961.13, due } year and 5 months hence, at 7 per cent. ?.
11. What is the discount of $37.40, due at the end of 7 months, at 6 per cent.?.
COMPOUND INTEREST. · 69. When at the end of cach year, the-interest due is added to the principal, and the amount thus obtained is considered as a new principal, upon which the interest is cast for another year, and added to it to form a new principal for the next year, and so on to the last year, the last amount thus obtained, is called the AMOUNT AT COMPOUND. INTEREST. If from this amount we subtract the original principal, we obtain the COMPOUND INTEREST.
Examples. ..1. What is the compound interest of $1000, for 3 years, at 7 per cent.? .- Principal,
á $1000 Interest on $1000 for one year,
70 First amount, or second principal,
1070 Interest on $1070 for one year,
Second amount, or third principal,
2. What is the compound interest of $100, for 4 years, at 6 per cent.? Principal, .
$100 Interest for first year,
First'amount, or second principal,
126.25 Original principal, Compound interest required,
Ans. $26.25 3. What is the compound interest of $630, for 4 years, at 5 per cent.?
Ans. $135,769. By carefully reviewing the above manner in which com. pound interest is computed, we discover that the successive amounts, which are considered as new principals, form the terms of a geometrical series, whose first term is the original principal, the ratio is the amount of $1, for one year, at the given rate per cent.; the number of terms is equal to the num. ber of years, plus one.
From this we learn, that finding the amount of a given prin. cipal, for a given number of years, at a given rate per cent., consists in finding the last term of a geometrical progression, when the first term, the ratio, and the number of terms are given. This question has been solved by Case I., of geometrical progression.
The following table gives the amount of $1, or £1, for any number of years, not exceeding 30, at 3, 4, 5, and 6 per cent., at compound interest, the interest being compounded yearly.
Years. 3 per cent. | 4 per cent. | 5 per cent. | 6 per cent.
1 1.030000 ) 1.040000 1.050000 / 1.060000
1.159274 | 1.216653 1.2762€2 1.338220
1.512590 1.731676 1.979932 2.260904
1.604706 1.872981 2.182875 2.540352 17 1.652848 1.947900 2.292018 2.692773
1.702433 2.025817 2.406619 2.854339
1.753506 2.106849 2.526950 3.021599
1.860295 2.278768 -2.785963 3.399564