9. What sum will amount to £387. 7s. 73d. 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.

166. 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. 163), 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. 162 and 163).

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,

£720
5

£36.00

£720 1st principal,

361st interest,

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

5

£37.80
20

16.00s.

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

.. £833. 9. 93= 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. 9s. 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 52 years), to find the compound interest for the 6th year, and then take ths of the last interest for the 2ths 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.

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 2 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. 138. 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 4 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.

167. A owes B £500, which is to be paid at the end of 9 months from the present time: now it is clear that, if the debt be discharged at once (interest being reckoned, we will suppose, at 4 per cent. per annum), Bought 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.

DEF. We may therefore define PRESENT WORTH to be the actual worth at the present time of a sum of money due some time hence, at a given rate of interest; and we may define the DISCOUNT of a sum of money to be the interest of the Present Worth of that sum, calculated from the present time to the time when the sum would be properly payable.

PRESENT WORTH.

168. RULE. "Find the interest of £100 for the given time at the given rate per cent., and state thus:

£100+ its interest for the given time at the given rate per cent. : given sum :: £100.: present worth required,"

Ex. 1. Find the present worth of £500, due 9 months hence, at 4 per cent. per annum.

Proceeding according to the above Rule,

Interest of £100 for 9 months at 4 per cent. is £3,

.. £103 : £500 :: £100: required present worth, whence, required present worth £485. 8s. 8d.

The reason for the above process is clear from the consideration, that £100 in 9 months at 4 per cent. interest would amount to £103, and therefore £100 is the present value of £103 due 9 months hence: and consequently we have

1st debt: 2nd debt :: 1st present worth: 2nd present worth.

Ex. 2. Find the present worth of £838, due 19 months hence, at 3 per cent. simple interest.

Since the interest of £100 for 19 months, at 3 per cent.

= £(}}×3)= £Y = £42,

.. £104: £838 :: £100: required present worth,

whence, required present worth £800.

Ex. 3. What is the value, at 16 years of age, of a legacy of £1000 payable at 21 years of age, allowing simple interest at 4 per cent.?

Since £100 at 4 per cent. simple interest will in 5 years amount to £120, therefore the present worth of £120 due years hence will at that

169.

RULE.

"Find the interest of £100 for the given time at the

given rate per cent., and state thus:

£100+ its interest for the given time at the given rate per cent. : given sum :: interest of £100 for the given time at the given rate per cent. : discount required."

Ex. 1. Find the discount of £500, due 9 months hence, at 4 per cent. per annum.

Proceeding according to the above Rule,

The interest of £100 for 9 months at 4 per cent. = £3; therefore proceeding according to the Rule,

£103 : £500 :: £3 : required discount,

whence, required discount = £14. 11s. 3d.

The reason for the above process is clear from the consideration, that £3 is the interest for 9 months, at 4 per cent., of £100, the present worth of £103 due at the end of that time; and consequently we have

1st debt: 2nd debt :: discount on 1st debt : discount on 2nd debt.

Ex. 2. Find the discount on £1000, due 15 months hence, at 5 per cent. per annum.

The interest of £100 for 15 months at 5 per cent. = £6. 5s. ;

.. £106. 5s.: £1000 :: £6. 5s. : required discount,

whence, required discount = £58. 16s. 511d.

Ex. 3. Find the discount on £127. 2s. for half a year at 5 per cent.

Note 1. Discount=given sum less Present Worth; Present Worth= given sum less Discount.

Note 2. In the discharge of a tradesman's bill it is usual to deduct interest instead of discount; thus, if B contracts with A a debt of £100, A giving 12 months' credit, it is usual in business, if the interest of money be reckoned at 5 per cent. per annum, and the bill be discharged at once, for A to throw off £5, or for A to receive £95 instead of £100; but if A were to put out the £95 at 5 per cent. interest it will not amount to £100 in 12 months; therefore such a proceeding is to the advantage of B: the sum of money which in strictness ought to have been deducted, was not £5, the interest on the whole debt, but £4. 15s. 2 d., the interest of the present worth of the debt, i. e. the discount.

Note 3. Bankers and Merchants in discounting bills calculate interest, instead of discount, on the sum drawn for in the bill, from the time of their discounting it to the time when it becomes due, adding THREE DAYS OF GRACE, which days are allowed in England after the time a bill is