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1 XT The interest of 1 for one year is or simply r, if r is

100 considered a decimal. The amount of 1 for one year then, will be 1 +r. The amount of p dollars will be p (i+r).

For the second year, p (1 + r) will be the principal, and the amount of 1 being (1+r), the amount of pli +r) will be p (1+r) (1 + r) or p (1.+ r).

For the third year p (1 + r)? being the principal, the amount will be p(1+r)? (1 + r) or p (1 + r)".

For n years then, the amount will be p(1+r)".
Putting A for the amount, we have

A=p (1 + r)* This equation contains four quantities, A, p, r, and n, any three of which being given, the other may be found.

Logarithms will save much labour in calculations of this kind.

Examples

1. What will $753.37 amount to in 5 years, at 6 per cent. compound interest ?

Here p=753.37, r=.06, and n = 51. log. 1 tr=1.06

0.025306

53

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2. What principal put at interest will amount to $5000 in 13 years at 5 per cent. compound interest?

By the above formula

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log. p=$2651.60 nearly Ans.

3.423513 3. At what rate per cent. must $378.57 be put at compound interest, that it may amount to $500 in 5 years ?

Solving the equation A=p (1 + r)" making the unknown quantity, it becomes

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Dividing by n = 5

0.120824 (5 log. (r + 1) = 1.05722

0.024165 Consequently r= 0.05722 Ans. 4. In what time will $284.37 amount to 750 at 7 per cent.?

Making n the unknown quantity, the equation A=P(1 + r)* becomes

A

=nx log. (1 + r), and P

log

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log. n = 14.334 nearly Ans.

1.156356 5. What will be the compound interest of $947 for 4 years and 3 months at 54 per cent. ?

6. What will $157.63 amount to in 17 years at 47 per cent.?

7. A note was given the 15th of March 1804, for $58.46, at the rate of 6 per cent. compound interest; and it was paid the 19th of Oct. 1823. To how much had it amounted ?

8. A note was given the 13th of Nov. 1807, for $456.33, and was paid the 23d of Sept. 1819. The sum paid was $894.40. What per cent. was allowed at compound interest ?

9. In what time will the principal p be doubled, or become 2 p, at 6 per cent. compound interest? In what time will it be tripled ?

Note. In order to solve the above question, put 2 p in the place of A for the first, 3 p for the second, and find the value

of n.

The principles of compound interest will apply to the following questions concerning the increase of population.

10. The number of the inhabitants of the United States in A. D. 1790 was 3,929,000, and in 1800, 5,306,000. What rate per cent. for the whole time was the increase ? What per cent. per year?

11. Suppose the rate of increase to remain the same for the next 10 years, what would be the number of inhabitants in 1810?

12. At the same rate, in what time would the number of inhabitants be doubled after 1800 ?

13. The number of inhabitants in 1810 by the census was 7,240,000. What was the annual rate of increase ?

14. At the above rate, what would be the number in 1820 ?

15. At the above rate, in what time would the number in 1810 be doubled ?

16. The number of inhabitants by the census of 1820, was 9,638,000. What was the annual rate of increase from 1810 to 1820 ?

17. At the same rate, what is the number in 1825 ?
18. At the same rate, what will be the number in 1830?

19. At the same rate, in what time will the number in 1820 be doubled ?

20. In what time will the number in 1820 be tripled ?

21. When will the number of inhabitants, by the rate of the last census, be 50,000,000 ?

LIII. 1. Suppose a man puts $10 a year into the savings bank for 15 years, and that the rate of interest which the bank is able to divide annually is 5 per cent. How much money will he have in the bank at the end of the 15th year? Suppose a = the sum put in annually,

r= the rate of interest,
t = the time,

A = the amount. According to the above rule of compound interest, the sum a at first deposited will amount to a (r + 1)"; that deposited the second year will amount to a (r + 1)!“, that deposited

A=

the third year will amount to a (r + 1)^?; that deposited the last year will amount to a (r + 1)'. Hence we have = a (r + 1)' + a (r + 1)e- ta(r + 1)es, .a (r+ 1) ....

a[(r + 1) + (r + 1)e-! + (r + 1)e-e....(r + 1)] But (x + 1)', (r + 1)-, &c. is a geometrical progression, whose largest term is (r + 1), the smallest r + 1, and the ratio r +1. The sum of this progression, Art. XLVII. is

(r +1) [(* + 1)—1]

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The same result may be obtained by another course of reasoning

The amount of the sum a for one year is a + ar. Adding a to this, it becomes 2 a ta r.

The amount of this at the end of another year is 2 a tar + 2 ar tam, or 2 at 3 ar + a ml. Adding a to this it becomes

3a + 3 ar tari. The amount of this for 1 year is 3a + 3 ar + aml + 3 ar + 3 a pe + am,

= 3a + 6ar + 4 ar + ar,

= a (3 + 6 r + 4 m2 to ne). This is the amount at the end of the third year before the addition is made to the capital. The law is now sufficiently manifest. With a little alteration, the quantity 3+ 6 r +433 + pd may be rendered the 4th power of 1+r. The three last coefficients are already right. If we add 1 to the quantity it becomes

4 + 6 + 4'm + r. Multiply this by r and it becomes

4r + 6 på t. 4 pk + r.

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