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.

153. RULE.

PRESENT WORTH.

"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;

.. if x=present worth required;

£103 £500 :: £100 x;

whence = £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, from the principle laid down in Art. (143), 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.

Let x=present worth required;

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

= £(1×3)= £Y = £43 ;

.. £104: £838 :: £100 x; whence x = £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.?

DISCOUNT.

154. 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,

Let x=discount required;

interest of £100 for 9 months at 4 per cent. = £3;

.. £103 £500 :: £3: x;

whence = £14. 118. 31d.

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.

Let x= discount;

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

.. £106. 5s.: £1000 :: £6. 5s. : x;

whence = £58. 16s. 514d.

Ex. 3. Find the discount on £127. 2s. for half-a-year at 5 per cent. Let x= discount required;

then £100 £127 :: £1⁄2: x; whence x=£3. 28.

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

155. Examples in Present Worth and Discount, at simple or compound interest, can be solved by the formulæ,

For if V and D represent present worth and discount respectively,

P, r, n,

.......... ..........

as in Arts (149 and 151).

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 NOMINALLY due, before it is LEGALLY due; which is of course an additional advantage. When a bill is payable on demand, the days of grace are not allowed.

Note 4. If a bill, without the days of grace, should appear to be due on the 31st of any month which contains only 30 days, the last day of that month, and not the first day of the next, is considered as the day on which the bill is due. Thus a bill drawn on the 31st of October, at 4 months, would be really due, adding in the days of grace, on the 3rd of March. Also bills which fall due on a Sunday, are paid in England on the previous Saturday.

Ex. A bill of £1000 is drawn on Feb. 16th, 1851, at 7 months' date; it is discounted on the 8th of July at 5 per cent. What does the banker gain by the transaction?

The bill is legally due on Sept. 19; and from July 8 to Sept. 19 are

Ex. LXI.

1. Find the Present Worth of

(1) £283. 10s. due 1 year hence, at 5 per cent. per annum, simple

(1) £63. 6s. 8d.

(2) £1380. 7s. 6d.

(3) £107.5s.

(4) £125. 10s.

due 4 months hence, at 4 per cent. per annum,

[simple interest.

(5) £487

(6) £340

(7) £3640

(8) £813. 9s.

(9) £250. 15s.

(10)

£55

146 days.........

43........

(11) A bill of £649 is dated on June 23, 1853, at 6 months, and is discounted on July 8, at 31 per cent.; what does the banker gain thereby?

(12) Find the true discount on a bill drawn March 17, 1853, at 3 months, and discounted May 2, at 53 per cent.

(13) Find the simple interest on £545 in 2 years, at 3 per cent. per annum; and the discount on £583. 3s. due 2 years hence,

at the same rate of interest. Explain clearly why these two sums are identical.

(14) Explain the difference between Discount and Interest.

Five volumes of a work can be bought for a certain sum, payable at the end of a year; and six volumes of the same work can be bought for the same sum in ready money: what is the rate of discount?

(15) A tradesman marks his goods with two prices, one for ready money, and the other for one year's credit allowing discount at 5 per cent.; if the credit price be marked at £2. 9s., what ought to be the cash price?

STOCKS.

156. If the 3 per cent. Consols be quoted in the money-market at 963, the meaning of this is, that for £96. 7s. 6d. of money a person can purchase £100 stock, for which he will receive an acknowledgment which will entitle him to half-yearly dividends from Government, at the rate of 3 per cent. per annum on the stock held by him.

Similarly, if shares in any trading company, which were originally fixed at any given amount, say £100 each, be advertised in the sharemarket at 86, the meaning of this is, that for £86 of money one share can be obtained, and the holder of such share will receive dividends at the end of each half-year upon the £100 share, according to the state of the finances of the company.

DEF. STOCK may therefore be defined to be the capital of trading companies; or to be the money borrowed by our or any other Government, at so much per cent., to defray the expenses of the nation.

The amount of debt owing by the Government is called the NATIONAL DEBT, or the FUNDS. The Funds represent the credit of the country, which is bound to pay whatever debts are contracted by its Government. The government, however, reserves to itself the option of paying off the principal at any future time whatever; pledging itself, nevertheless, to pay the interest on it regularly at fixed periods, in the mean time.

From a variety of causes the price of stock is continually varying. A fundholder can at any time convert his stock into money, and it will depend upon the price at which he disposes of his stock, as compared with