PROBLEMS IN COMPOUND INTEREST.

WRITTEN EXERCISES.

493. To find the principal, when the compound interest, the time, and the rate are given.

1. What principal at 6% compound interest will produce $2372.544 interest in 10 years?

$1.790848-$1-$.790848.

$2372.544.790848

EXPLANATION. -By geomet

$3000.

rical progression, or by the compound interest table on page

272, the amount of $1 at compound interest for the given time at the given rate is found to be $1.790848.

That sum less $1 gives $.790848, the compound interest of $1 for the given time at the given rate. Then, $2372.544.790848 $3000, the principal.

2. What principal at 6% compound interest will produce $3150 interest in 8 years?

3. What principal at 5% compound interest will produce $2896 interest in 12 years?

4. What principal at 7% compound interest will produce $3600 interest in 15 years? At 4% in 20 years?

494. To find the rate, when the principal, compound interest, and time are given.

1. At what rate per cent will $500 yield $203.55 compound interest in 7 years?

$203.55÷500-$.4071.

EXPLANATION.-Since $203.55 is the compound interest of $500 for 7 years,

of that sum will be the compound interest of $1 for the same time. By referring to the compound interest table, opposite 7 years, we find the amount $1.4071, or the interest $.4071, in the 5% column. Therefore, the rate is 5%.

2. At what rate per cent will $1000 yield $ 503.63 compound interest in 7 years?

3. At what rate per cent will $1200 yield $721.2384 compound interest in 12 years?

4. What is the rate per cent when $1800 yields $ 901.314 compound interest in 6 years?

5. What is the rate per cent when $2000 yields $4344.338 compound interest in 15 years?

495. To find the time when the principal, the compound interest, and rate are given.

1. In what time will $600 amount to $1200 at 7% compound interest?

amount to of $ 1200, or $2.

By the compound interest table, $1 at 7% will in 10 yr.

amount to $1.967151, and in 11 yr. to $2.104852, consequently the time must be between 10 and 11 yr. The interest of $1.967151 for a year at 7% is $.137701, and the difference between $2 and $1.967151 is $.032849. Since the interest of $1.967151 for a year is $.137701, to earn $.032849 will require 82849 of a year, or 2 mo. 26 da. Therefore the time is 10 yr. 2 mo. 26 da.

2. In what time will $400 amount to $1000, at 6% compound interest?

3. In what time will $750 amount to $1500, at 5% compound interest? Or, in how long a time will any sum double itself at 5% ?

4. In what time will $960 amount to $2000, at 7% compound interest?

5. In what time will $1300 amount to $2500, at 6% compound interest?

6. In what time will $3200 amount to $4800, at 4% compound interest ?

ANNUITIES.

496. A definite sum of money payable at the end of equal periods of time is an Annuity.

Properly speaking, an annuity is a sum payable annually, but sums payable at intervals of quarter-years, half-years, or other periods, are also called annuities.

497. An annuity which continues forever is called a Perpetual Annuity or Perpetuity.

498. An annuity which commences at a definite time, and continues for a definite time, is called a Certain Annuity.

499. An annuity whose commencement or continuance, or both, depend upon some contingent event, as the death of some person, is called a Contingent Annuity.

500. An annuity upon which the payments were not made when they were due is called an Annuity in Arrears or Forborne.

501. The Amount or Final Value of an annuity is the sum of all the payments, increased by the interest of each payment, from the time it becomes due until the annuity ceases

502. A sum of money, which, upon being put at interest for the given time at the given rate, will be equal to the amount of the annuity, is the Present Value of the annuity.

503. Annuities are sometimes computed at Simple Interest and sometimes at Compound Interest.

WRITTEN EXERCISES.

504. Annuities at Simple Interest.

1. What is the amount of an annuity of $500, unpaid for

til the time of payment, or 4 yr.; the second payment will draw interest for 3 yr.; the third payment, for 2 yr.; the fourth payment, for 1 yr. Hence, these sums form an arithmetical progression, the first term of which is $ 500, the common difference, the interest of $500 for 1 yr., or $30, and the number of terms, 5. The sum of this series will be the amount due.

2. What is the present worth of an annuity of $2500 to remain unpaid for 6 years, interest at 6% ?

$ 17,250 = Amount of annuity. $17,250 ÷ 1.36 = $12,683.82

EXPLANATION.-The amount due at the end of 6 years would be $17,250.

Since this sum is not payable

for 6 years, the present worth of it is found by dividing by the amount of $1 for the given time at the given rate.

3. What is the amount of an annuity of $800, unpaid for 4 years, at 6% ?

4. What is the amount of an annuity of $960, payable semi-annually, at 6%, but unpaid for 4 years?

5. What is the present worth of an annuity of $1500, to remain unpaid for 8 years, at 6%?

6. Mr. L. has an annuity of $1800, payable quarterly. If it remains unpaid for 3 years 9 months, what will be the amount due at 8 % ?

7. A house was rented for $45 a month for 21 years. What sum would pay the entire rent in advance if it was not due until the lease expired, interest at 6% ?

ANNUITIES AT COMPOUND INTEREST. 385

505. Annuities at Compound Interest.

1. What is the amount of an annuity of $200 which is 20 years in arrears, compound interest at 6% ?

$200 × (1.0620—1). 1.06-1

=$7357.12

EXPLANATION. The payment now due is $200; the payment 1 year in arrears is $200 × 1.06; the payment 2 years in arrears is $200

× 1.06 × 1.06, or $200 × 1.062; the payment 3 years in arrears is $200 × 1.063. Thus it appears that the sums unpaid form a geometrical series, of which the first term is $200, the ratio 1.06, and the number of terms 20. The sum of this series is $7357.12, the amount of the annuity.

1. In finding the value of 1.0620, or similar expressions, use the compound interest table on page 272.

2. The amount of an annuity at simple interest is the sum of an arithmetical series; the amount of an annuity at compound interest is the sum of a geometrical series.

2. What is the amount of an annuity of $225, which is 6 years in arrears, compound interest at 7% ?

3. What is the amount of an annuity of $300, which is 9 years in arrears, compound interest at 6% ?

4. What is the amount of an annuity of $ 450, in arrears for 15 years, compound interest at 5% ?

5. What is the amount of an annuity of $650, in arrears for 10 years, the interest being compounded semi-annually, at 6% ?

6. A young man spends $50 a year for tobacco. What will this amount to in 20 years, at 6% compound interest?

7. What is the present value of an annuity of $800 for 6 years, compound interest at 5% ?

SUGGESTION. - Divide the amount of the annuity by the amount of $1, at compound interest, for the given time and rate.

8. What is the present worth of an annuity of $480 for 12 years, compound interest at 6% ?

9. A man purchased an annuity of $600 a year for 15 years, at 6% compound interest. What did it cost him?

STAND. AR. - 25