The following examples may be worked by the aid of the foregoing table:

7. What is the present worth, interest being 7 per cent., of $320, due at the end of 6 months and 3 days?

8. What is the discount on $750, due 9 months hence, at 7 per cent.?

Ans. $37.411.

9. What is the present worth of $3471.20, due 3 months and 9 days hence, at 7 per cent.?

Ans. $3405 643. 10. What is the discount of $150, due 3 months and

18 days hence, at 7 per cent.?

Ans. $3.085.

11. What is the discount of $961.13, due 10 months

67. SUPPOSE A owes me $100, due at the end of 3 months, and $100, due at the end of 9 months, and I wish him to give me one note of $200, of such a time that its present value shall be the same as the sum of the present values of the two individual debts. long after date must this note be made payable?

How

By the foregoing table for the present worth, we find the present value of $100 for three months, to be $98-2801: the present value of $100 for 9 months, to be $95.0119. Taking the sum, we have $193.292 for the present value of $200, due at a future time, which time we are required to find.

We may obviously consider $193-292 as a principal, which, at the given rate per cent., will amount to $200, in the time sought. The interest is, therefore, $6708.

This question now is equivalent to the following:

Given the principal, the rate per cent., and the interest, to find the time.

This has been solved under Prob. IV., Art. 65. Proceeding according to this rule, we find the interest of $193.292 for 1 year, at 7 per cent., to be $13:53, nearly. Dividing $6.708 by $13.53, we get 0:4958 of a year, equal to 5 months, 28-49 days, nearly. The equated time, when found by the ordinary method, is 6 months.

From the above, we deduce the following rule for the equation of payments, founded on the principle of equivalent present values.

RULE.

Find the sum of the present values of the individual debts; also, the sum of their discounts. Then regard the sum of the present values of the individual debts as a principal, and the sum of their discounts as the interest. Then proceed with this principal and interest and given rate per cent., according to Rule under Prob. IV., Art. 65.

EXAMPLE.

Suppose A owes me $500, due at the end of 3 months, $600 at the end of 4 months, and $800 at the end of 6 months. How long may the whole $1900 remain unpaid, so that its present worth may be the same as the sum of the present worths of the individual debts?

Present value of $500, due 3 months hence, is $191.4005

$1850-6671

1850-6671

Sum of their present values is

Sum of several payments $1900·0000

Sum of their discounts is $49.3329

Now, if a principal of $1850-6671 gives $49.3329 interest at 7 per cent., what is the number of years ?

By rule under Prob. IV., we find that the interest on $1850-6671 for 1 year, at 7 per cent., to be $129.5467. Hence, dividing $49-3329 by $129-5467, we find 0-3808 of a year, which is the same as 4 months, 17·088 days, nearly.

If we find the equated time by the usual rule for equation of payments, it will be 4 months, 17:368 days, nearly, which differs less than half a day from the result by the above method.

Other examples might be given to illustrate this method of finding the time for the equation of payments, on the principle of equivalent present values, but enough has been done to call the attention of the student to this singular subject.

CHAPTER VII,

COMPOUND INTEREST.

68. WHEN, at the end of each 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 If from this amount we subtract the original principal, we obtain the COMPOUND INterest.

COMPOUND INTEREST.

EXAMPLES.

1. What is the compound interest of $1000 for 3 years, at 7 per cent.?

The compound interest required, Ans. $225 043

2. What is the compound interest of $100 for 4 years,

3. What is the compound interest of $630 for 4 years, Ans. $135 769.

at 5 per cent.?

By carefully reviewing the above manner in which compound 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 number of years, plus one.

From this we learn that finding the amount of a given principal, 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.

Thus, the amount of $1 for one year, at 3 per cent., is $1.03; for two years, it is $(103); for three years, it is $(103)3; for four years, it is $(1·03)a; and in general, for any number of years, it is found by raising 103 to a power denoted by the number of years.