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Find the proceeds, if this note was discounted at the bank, at 8%, May 29, 1903.

9. My note of $360, which will be legally due in 1 yr. 4 mo. 20 da., is discounted at a bank, at 8%. What rate of interest does the banker receive?

NOTE.—The difference between true discount and bank discount is equal to the simple interest on the true discount for the given time at the given rate. Illustrate this fact by solving the following example:

10. Find the difference between the true discount and the bank discount on $440, due in 1 yr. 8 mo., at 6%, not reckoning days of grace.

III. ANNUAL INTEREST

350. Annual interest is simple interest on the principal, augmented by the simple interest on each interval's interest from the close of the interval to the time of settlement.

EXAMPLE A note of $600.00 drawing 6% interest, payable annually, runs for 4 years and 3 months. Find the amount due at maturity, if no payments have been made.

SOLUTION: Each year's interest will bear interest after it becomes due, as follows:

1. 1st year's interest for 3 years, 3 months.

2. 2d year's interest for 2 years, 3 months. I.

3. 3d year's interest for 1 year, 3 months.
4. 4th year's interest for 0 years, 3 months.

Total
To the simple interest on the principal for 4 years, 3

7 years.

months, we must add the interest on 1 year's interest for the sum of the above intervals, 7 years.

1. Int. on $600 for 1 yr. at 6% $36, due annually.

2. Int. on $600 for 41 yr. at 6% = $153, int. on prin. II.

3. Int. on $36 for 19 yr. at 6% $15.12, int. on int.
4. $600 + $153 + $15.12 $768.12, amount due.

Exercise XLII

1. A note of $350 drawing 5% interest, payable annually, runs 4 years, 6 months. Find the amount due at maturity, if no payments have been made.

2. A man bought $500 worth of goods on 9 months' credit; he paid for them at the end of 3 years, 3 months. Allowing 6% interest, payable annually, how much was due?

3. A gentleman holds six $1000 railroad bonds, due in 3 years, interest 6% payable semi-annually: no interest having been paid, what amount is owing him when the bonds mature?-R. N. H., p. 260.

NOTE.-Frequently annual or periodic interest is payable semiannually or quarterly.

4. Find the annual interest on $800 for 1 year, 9 months, at 5% per annum, payable quarterly.

5. The annual interest on a sum of money for 4 years, 6 months, at 5%, paid annually, was $85.75. What was the sum?

SUGGESTION.—Find the annual interest on $1.

6. Find the amount due at the end of 5 years, 9 months,

of a note of $1200, bearing interest at 6%, payable annually, if no payments have been made.

REMARK.-In some states, the law provides that unpaid annual interest shall bear interest at the legal rate. In the following problem, the unpaid interest bears the legal rate of interest.

7. In a state whose legal rate is 6%, H makes a note of $1250 for 3 yr. 3 mo., with interest at 8%, payable semiannually; he pays no interest; find the amount due at maturity.-S. & K., p. 197.

IV. COMPOUND INTEREST

.05

351. Compound interest is the interest that accrues by making the interest, due at the close of any interval, a part of the interest-bearing principal for the next succeeding interval.

EXAMPLE Find the compound interest on $1 for 3 years at 5%, payable annually.

SOLUTION :
$1.00 = principal (P). .

rate (R).
$0.05

int. for 1st yr.
1.00
$1.05 (1 + R) = amount for 1st yr. (A).

.05
$0.0525 int. for 2d

yr. .
1.05
$1.1025 (1 + R)2 = amount for 2d

yr.
.05
$0.055125

int. for 3d yr.
1.1025
$1.157625 (1 + R): amount for 3d yr.

1. = the original principal.
$0.157625 the compound int. for 3 years.

A careful study of this example will show that the compound amount of $1, at any rate, for any number of years, equals (1 + R) raised to the power indicated by the number of years. Then to get the amount of n dollars, take n times the amount of $1. Using the initial letters, we have

we have the foilowing formulas:

I. A = P(1 + R)T

II. P

A
(1 + RT

III. I= A - P

Write the rule corresponding to each formula.

352. The interest interval is generally a year, half-year, or quarter-year. The length of the interval is indicated by inserting annually, semi-annually, or quarterly.

The half-interval rate V1 + interval rate – 1. Thus, if the annual rate is 21%, the semi-annual rate would V1.21 – 1 = 1.10 – 1 = .10, or 10%.

This fact may be illustrated thus:

1. Am't of $100 for 1 yr. at 21% annually = 1.21 x $100 = $121. 2. Am't at 21% semi-annually = V1.21 x V1.21 x $100 = $121. 3. .. the semi-annual multiplier = V1.21, and the rate= V1.21–1.

Similarly, it can be shown that the quarterly multiplier

V1.21, and the rate V1.21 – 1.

TABLE Showing the amount of $1 at compound interest from 1 year to

20 years.

Yr. 124 per cent. 3 per cent. 31 per cent. 4 per cent.

[blocks in formation]

1 1.025

1.03 1.035 1.04 1.05 1.06 2 1.050625 | 1.0609 1.071225 1.0816 1.1025 1.1236 3 1.076891 1.092727 | 1.108718 1.124864 | 1.157625 1.191016 4 1.103813 1.125509 1.147523 | 1.169859 | 1.215506 | 1.262477 5 1.131408 1.159274 1.187686 1.216653 1.276282 | 1.338226 6 1.159693 | 1.194052 | 1.229255 | 1.265319 | 1.340096 1.418519 7 1.188686 1.229874 1.272279 | 1.315932 1.4071 1.50363 8 1.218403 1.26677 1.316809 | 1.368569 | 1.477455 | 1.593848 9 1.248863 | 1.304773 | 1.362897 1.423312 | 1.551328 1.689479 10 1.280085 | 1.343916 1.410599 1.480244 1.628895 1.790848 11 1.312087 | 1.384234 | 1.45997 1.539454 | 1.710339 1.898299 12 1.344889 1.425761 1.511069 1.601032 1.795856 2.012197 13 1.378511 1.468534 | 1.563956 1.665074 | 1.885649 2.132928 14 1.412974 | 1.51259 1.618695 | 1.731676 1.979932 2.260904 15 1.448298 1.557967 1.675349 1.800944 2.078928 2.396558 16 1.484506 | 1.604706 1.733986 1.872981 2.182875 2.540352 17 1.521618 1.652848 1.794676 | 1.947901 2.292018 2.692773 18 1.559659 1.702433 | 1.857489 2.025817 2.406619 | 2.854339

1.59865 1.753506 1.922501 2.106849 | 2.52695 3.0256 20 | 1.638616 1.806111 | 1.989789 2.191123 2.653298 3.207136

19

Yr.

.

7 per cent.

8 per cent.

9 per cent. 10 per cent. 11 per cent. 12 per cent.

1 1.07 1.08

1.09

1.10 1.11 1.12 2 1.1449 1.1664 1.1881 1.21

1.2321 1.2544 3 1.225043 | 1.259712 | 1.295029 | 1.331 1.367631 1.404908 4 1.310796 | 1.360489 1.411582 | 1.4641 1.51807 1.573519

1.402552 | 1.469328 | 1.538624 1.61051 1.685058 | 1.762342 6 1.50073 1.586874 | 1.6771 1.771561 1.870414 1.973822 7 1.605781 1.713824 | 1.828039 1.948717 2.07616 2.210681 8 1.718186 1.85093 1.992563 2.143589 2.304537 | 2.475963 9 1.838459 1.999005 | 2.171893 2.357948 | 2.558036 | 2.773078 10 1.967151 | 2.158925 2.367364 2.593742 2.83942 3.105848 11 2.104852 2.331639 2.580426 2.853117 / 3.151757 | 3.478549 12 2.252192 2.51817 2.812665 3.138428 3.49845 3.895975 13 2.409845 | 2.719624 3.065805 3.452271 | 3.883279 4.363492 14 2.578534 2.937194 | 3.341727 | 3.797498 | 4.31044 4.887111 15 2.759031 3.172169 3.642482 | 4.177248 4.784588 5.473565 16 2.952164 | 3.425943 3.970306 4.594973 5.310893 6.130392 17 3.158815 3.700018 4.327633 5.05447 5.895091 6.86604 18 3.379932 3.996019 4.71712 5.559917 | 6.543551 | 7.689964 19 3.616527 | 4.315701 5.141661 6.115909 7.263342 8.61276 20 3.869684 4.660957 5.604411 6.7275 8.062309 | 9.646291

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