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17. As a body cools, at any time t the difference in temperature 0 between the body and the surrounding medium is given by the equation 0 = Ope-mt. If a fireless cooker oven is 430° above room temperature when t 0, and 216° after one hour, how long before it will be 10° above room temperature?

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86. Compound Interest. Interest is the money paid for the use of borrowed capital which is called the principal. Interest that is paid only on the principal is called simple interest. When interest is paid on unpaid interest, which is added to the principal periodically, it is called compound interest. If unpaid interest is added to the capital at yearly intervals it is said to be compounded annually. If the interest is added every six months or three months it is said to be compounded semiannually or quarterly.

In any problem in compound interest where the interest is compounded annually, four quantities are involved:

P, the principal;

i, the rate of interest, which is the sum of money in dollars paid for the use of one dollar for one year. Thus, if the rate

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S, the amount, which is the sum of the principal and interest

for n years.

Theorem 1. The amount, S, of a principal of P dollars, interest compounded annually for n years at the rate i, is

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The amount of one dollar for one year at the rate i is 1 + i. Hence the amount of P dollars for one year is P(1 + i). The amount at the end of the second year is obtained by multiplying the new principal, P(1 + i), by the amount of one dollar for one year 1 + i.

Hence the amount at the end of the second year is P(1 + i)(1 + i) = P(1 + i)2.

Similarly, the amount at the end of the third year is P(1 + i)2(1 + i) = P(1 + i)3. Comparing these results, we see that the exponent of (1+i) is the same as the number of years, and hence

The amount S at the end of n years is S

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This equation involves four variables. If any three are given, the fourth may be determined by substituting the given values and solving for the unknown. The value of S, P, or i may be determined by algebraic processes while the determination of n involves the solution of an exponential equation. At what rate will $50 amount to $75 in 7 years? S = 75, and n = 7. 75 = 50 (1 + i)7,

EXAMPLE 1.

Here P = 50,

Hence

Then

So that

Hence

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log 1.5

=

0.1761.

log 1.5 0.0252.
V1.5 = 1.0598.

i = 0.0598, so that the rate is 5.98 %.

If the interest is compounded semi-annually (as in some savings banks), in n years there are 2n periods and the interest on one dollar for each period is i/2, so that the amount in n years is

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2n

If the interest is compounded quarterly, the amount in n years is

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Theorem 2. If interest is compounded m times a year, at the annual rate i, the amount after n years is the same as if the interest were compounded annually at the rate i/m, for mn years. That is

S = P(1 + i/m)mn.

EXAMPLE 2. Find the amount of $50 for 10 years with interest convertible into principal semi-annually at 4%.

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DEFINITION. The present value, P, of a sum of money, S,

due after n years, is the principal which must be placed at

compound interest at a given rate i, in order to amount to the sum S at the end of n years.

Thus in the formula in Theorem 1, P is the present value of S.

EXERCISES

1. A principal of $100, deposited in a trust company, bears interest at the rate of 4% compounded semi-annually. What will the balance be at the end of ten years if no withdrawals are made?

2. A mother promises her twelve year old boy that she will present him with $100 on his twenty-first birthday provided he abstains from smoking until that time. How much should she deposit in the trust company of the previous example in order to have an amount of $100 when the boy is twenty-one?

3. At what rate would a sum of money double itself in 25 years if interest is compounded annually?

4. In how many years will a sum of money double itself if placed at compound interest if (a) the rate of interest is 5%, compounded annually? (b) the rate is 4% compounded semi-annually?

5. If $100 be deposited in a trust company at 4%, compounded semiannually, how long before it will amount to at least $150?

6. A building and loan association offers an opportunity for an investment to yield 8%, compounded quarterly.

(a) If $100 is invested, to what will it amount in 5 years?

(b) What sum should be invested to amount to $120 in 7 years? (c) How long must $50 be invested to amount to $175?

(d) How long will it take any principal to double itself?

(e) How long will it take for a sum to treble itself?

7. A man borrows $50 from a friend, and three years later returns $65. What is the equivalent rate of interest, if interest is compounded quarterly? 8. A man bought a diamond for $150 in 1900, and sold it for $400 in 1915. What rate of interest did he realize, assuming it to be compounded annually?

9. If a building lot is bought for $500, and its value increases by 10% annually, what will it be worth in 5 years?

10. The number of students in a college increased from 275 to 460 in 10 years. At what rate did the number increase, assuming that the percentage of increase each year was constant?

11. What sum should be deposited in a bank paying 3% compounded semi-annually in order to pay off a debt of $500 due three years later? 12. Construct the graph of the amount of one dollar, interest compounded annually at 6%, as a function of the number of years n. Build the table of values for n = 10, 20, 30, 40.

13. Verify the fact that the values of S obtained in Exercise 12 are In geometrical progression.

87. Annuities. DEFINITION. An annuity consists of a series of equal payments made at equal intervals of time. The first payment of an annuity is made at the end, not at the beginning, of the first period of time, unless otherwise specified.

Annuities are common in commercial life. The rent of a house or store, the premium of an insurance policy, the dividend on a bond, wages and salaries, the payments of interest on a mortgage, are examples of annuities.

Theorem 1. The amount of an annuity of R dollars, payable annually, accumulated for n years at the rate i, interest compounded annually is,

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The compound interest on each payment is obtained by the method of the preceding section.

1 years and amounts

Since the first payment is made at the end of the first year, the first payment is at interest for n to R(1 + i)-1.

The second payment is at interest for n 2 years and amounts to R(1 + i)-2, etc.

The next payment to the last is at interest 1 year and amounts to R(1 + i).

The last payment bears no interest and amounts to R.
Adding these amounts in the reverse order we obtain

K = R + R(1 + i) + . . . + R(1 + i)n−2 + R(1 + i)n−1. The terms on the right form a geometric progression whose sum is obtained by the formula where the first term

rl a

S

=

r 1'

is a R, the last term is l = R(1 + i)n−1, and the ratio is

=

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Theorem 2. The amount of an annuity, if R dollars are paid m times a year, accumulated for n years at the rate i, interest compounded m times a year, is equivalent to the amount of an annuity of R dollars, payable annually, accumulated for mn years at the rate i/m, interest compounded annually.

DEFINITION. The present value of an annuity of R dollars a year for n years is the amount that must be deposited in a bank, interest compounded annually at the rate i, so that, if R dollars be withdrawn annually, there will be no balance left after the nth withdrawal.

Theorem 3. The present value of an annuity of R dollars payable annually for n years is

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The present value of the annuity is the sum of the present values of R dollars due 1 year hence, R dollars due 2 years hence, etc., for n years. We have from the formula for compound interest that the present value P of a sum S due in n years is PS(1+i)". Hence if A denotes the present value of the annuity, we have

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A = R(1 + 2)−1 + R(1 + i)−2 + · · · + R(1 + 2)−n. The sum on the right is a geometric progression for which a = R(1 + i)−1, l = R(1 + i)", and r = (1 + i)−1.

R(1+i)",

Substi

tuting these values in the formula for the sum of a geometric progression, we have

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Multiplying the numerator and denominator of the right-hand side by (1 + i)+1 and simplifying, we have

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Theorem 4. The present value of an annuity, if R dollars are paid m times a year for n years, at the rate i compounded m times a year, is equivalent to that of an annual annuity of R dollars for mn years at the rate i /m compounded annually.

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