5. In what time will £142. 10s. amount to £227. 5s. 9d. at 3 per cent. simple interest?

6. At what rate will £157. 15s. 4d. amount to £295. 16s. 3d. in 25 years at simple interest?

7. What sum will produce for interest £56. 14s. in 21 years at 41 per cent. simple interest?

8. What sum will amount to £105. 6s. Od. in 3 years at 4 per cent., simple interest?

9. What sum will amount to £387. 78. 7 d. in 3 years at 4 per cent., simple interest?

10. In what time will £1275 amount to £1549. 11s. at 3 per cent., simple interest?

11. At what rate per cent., simple interest, will £936. 13s. 4d. amount to £1157. 78. 41d., in 47 years?

12. In what time will £125 double itself at 5 per cent. simple interest?

13. What sum will amount to £425. 19s. 4td. in 10 years at 3 per cent. simple interest, and in how many more years will it amount to £453. 11s. 7d.?

14. What sum of principal money, lent out at 5 per cent. per annum, simple interest, will produce in 4 years the same amount of interest as £250, lent out at 3 per cent. per annum, will produce in 6 years?

COMPOUND INTEREST.

150. To find the Compound Interest of a given sum of money at a given rate per cent. for any number of years.

RULE. "At the end of each year add the interest of that year, found by Art. (146), to the principal at the beginning of it; this will be the principal for the next year; proceed in the same way as far as may be required by the question. Add together the interests so arising in the several years, and the result will be the compound interest for the given period."

The reason for the above Rule is clear from what has been stated in Arts. (145-146).

Ex. Required the compound interest and the amount of £720 for 3 years at 5 per cent.

Proceeding as in Simple Interest for the 1st year;

by addition,

£756=2nd principal, of which find interest at 5 per cent.

£793. 16=3rd principal, of which find interest as above

5

£39.69. 0

20

13.808.

12

9'60d.

£793. 16

= principal for 3rd year,

39 13.9% interest for 3rd year,

.. £833. 98. 9 d. = amount of £720 in 3 years at 5 per cent.

compound interest.

The compound interest for that time

=sum of interests for each year,

= £36+£37. 16s. + £39. 13s. 93d.,

= £113. 98. 93d.

Note 1. It is customary, if the compound interest be required for any number of entire years and a part of a year, (for instance for 5 years), to find the compound interest for the 6th year, and then take ths of the last interest for the ths of the 6th year.

Note 2. If the interest be payable half-yearly, or quarterly, it is clear that the compound interest of a given sum for a given time will be greater

as the length of each given period is less; the simple interest will not be affected by the length of each period.

Note 3. As the vulgar fractions often in Compound Interest give considerable trouble, any sum in this Rule may be worked by means of decimals thus ;

Ex. Find the amount of £625 at the end of 3 years at 4 per cent. compound interest.

£.

625 Principal for 1st year

4.5

3125

2500

£28.125 Int. for 1st year

£625

£653-125 Principal for 2nd year

4.5

3265625

2612500

£29.390625 Int. for 2nd year £653.125

£682-515625 Principal for 3rd year

4:5

3412578125

2730062500

£30-713203125 Int. for 3rd year

£682.515625

£713 228828125

20

8.4 576562500

12

d.6.9187500

4

q.3.67500

.. Amount = £713. 48. 63d. 279.

151. Any example in Compound Interest may be worked by remembering the formula M= P(1+r)", which I now proceed to explain. Let P denote the principal (in pounds),

Since at the end of the first year, £1 amounts to (£1+r), and since the same proportion holds for each successive year, we obtain

£1 : (£1+r) :: £P : £P (1+r),

or the amount of £P in 1 year is £P (1+r).

Similarly

£1: (£1+r) :: £P (1+r) : £P (1+r) (1+r), or £P(1+r)2,

or the amount of £P in 2 years is £P (1+r)2.

Similarly, the amount of £P in n years is £P(1+r)”, or M= P(1+r)".

In which equation, any three of the quantities M, P, r, n being given, the fourth may be found.

Note. If the interest be payable half-yearly, we must put r for r, and 2n for n in the above formula.

Ex. Find the amount of £200 in 2 years at 4 per cent. compound interest.

Here

P=£200, n=2, r=0·04;
.. M= P(1+r)" = £200 (1·04)2,
= £216. 68. 4 d.

Ex. LX.

1. Find the compound interest of £2000 in 2 years at 4 per cent. per annum.

2. Find the amount of £800 in 3 years at 32 per cent., allowing compound interest.

3. Find the compound interest of £270 in 2 years, at 3 per cent.

4. Find the amount of £690 for 3 years at 4 per cent., compound interest.

5. Find the amount of £230. 15s. for 3 years, at 5 per cent., compound interest.

6. Find the difference in the amount of £415. 10s., put out for 4 years at 21 per cent., 1st at simple, 2nd at compound interest.

7. Find the compound interest of £130 in 3 years at 4 per cent. (interest being payable half-yearly).

8. What will £1760. 10s. amount to in 21 years, allowing 4 per cent. compound interest?

9. A person lays by £230 at the end of each year, and employs the money at 3 per cent. compound interest; what will he be worth at the end of 3 years?

10. Find the difference between the simple and compound interest of £416. 13s. 4d. for 2 years at 21 per cent.

11. What is the difference between the simple and the compound interest of £13,333. 6s. 8d. for 5 years, at 5 per cent.?

12. Find the amount of £180 in 3 years at 4 per cent. compound interest.

13. What sum of money put out to compound interest for 2 years at 5 per cent. will amount to £100?

14. What sum at 5 per cent. compound interest will amount in 2 years to £264. 12s.?

15. A and B each lend £256 for 3 years at 41 per cent. per annum, one at simple interest, the other at compound interest: find the difference in the amount of interest they respectively receive.

PRESENT WORTH AND DISCOUNT.

152. A owes B £500, which is to be paid at the end of 9 months from the present time: it is clear that, if the debt be discharged at once (interest being reckoned, we will suppose, at 4 per cent. per annum), B ought to receive a less sum of money than £500; in fact such a sum of money as will, being now put out at 4 per cent. interest, amount to £500 at the end of 9 months. The sum which B ought to receive now is called the Present Worth of the £500 due 9 months hence, and the sum to be deducted from the £500, in consequence of immediate payment, which is in fact the interest of the Present Worth, is called the Discount of the £500 discharged 9 months before it is due.