(15.) Suppose that the 3 per cents. consols sell for 70z1. per cent, when the 3 per cents. reduced sell for 71 per cent.; for instance, on January 20, which fund will be the most advantageous to purchase in, the interest on the consols being due the 5th January, and on the 3 per cents. reduced on the 5th of April?

(16.) If I buy 10,000l. capital in the India stock, in January, immediately after the dividends have been received at 20947. per cent. what will it cost me, allowing the broker per cent. on the capital for buying? and what do I make per cent. of my money, India stock bearing 10 per cent.

(17.) Suppose I have 6007. what nominal sum, in the Navy 5 per cents. will that purchase, at 10347. per cent. allowing the broker per cent. on the capital, or sum purchased?

(18.) On June 8th, 1818, I sold out 10007. consols. at 774, and, with the sum received, purchased in the Navy 5 per cents. at 106; what is my annual gain, in point of interest, my broker being allowed per cent, on the capital, in each transaction?

(19.) Which is the most advantageous, with respect to annual income, land bought at 25 years purchase *, or Bank-stock bought at 1687. per cent. the Bank-stock bearing 7 per cent. interest?

stock should sell for 2311. per cent. and that each will produce 41. 10s. 10d. per cent. interest, and therefore they are equally advantageous. Now this would be true, were the interests payable at the same time; but as that is not the case, the whole table, and all similar tables appear to be founded on error, and can tend only to mislead the public. To illustrate this remark-in the example before us, the India stock has a quarter's interest due on it at the time of purchasing, and therefore, in point of interest, is preferable to the Bankstock; and had the purchase been made in February at the same rate, the Bank stock would have had the advantage,

* Divide 1001. by the rate per cent., and the quotient will give the number of years purchase; that is, the number of years in which an estate will bring in the purchase-money: divide 1001. by the number of years purchase, and the quotient will give the rate per

cent.

DISCOUNT.

Definition. Discount, or Rebate, is an allowance made for the payment of any sum of money before it becomes due and the present worth of any sum, or debt, is such a sum as, if put to interest for the time, and at the rate for which the discount is to be made, would amount to the sum, or debt, due.

Proposition. Any sum, due some time hence, being given to find its present value to the creditor, discounting at any rate per cent.

Rule. As the amount of 1007. for the given rate and time, is to 1007. so is the given sum to its present worth. The difference between the given sum and its present value will give the discount.

Or, as the amount of 1007. for the given rate and time, is to the interest of 1001. for that time, so is the given sum to the discount. The difference between the given sum and its discount will give the present value.

Note. The preceding rule is built upon this basis, viz. that the present worth of any sum of money, due some time hence, put to interest for the time, for which the discount is to be made, should amount to the sum, or debt, due: and that the discount, put to interest for the same time, should amount to the interest of the sum due for that time,

2. Thus, the present worth of 1001. due one year hence, discounting at the rate of 5 per cent., is 951, 4s. 94d., and the discount of 1001. for one year, at the rate of 5 per cent., is 41. 15s. 2 d., according to the rule. Now, if the creditor should put the present money allowed him (viz. 931. 4s. 94d.) to interest, at the rate of 5 per cent. for one year, it will amount to 1001. exactly, and therefore he is not injured : again, if the debtor puts the discount allowed him (viz. 41. 15s. 2§d.) to interest, at the rate of 5 per cent. for one year, it will amount to 51., the exact sum which he might have made of the 1001, had he kept it in his hands till it became due.

3. When goods are sold to any amount, payable at different times, at the same or different rates per cent., calculate the present worth of each payment separately, as a debt independent of the other payments, and the sum of these will be the present value of the goods to the seller.

4. It is cnstomary with bankers and merchants, in discounting bills, to calculate the interest of the sum drawn for in the bill, from the, time of their discounting it to the time it becomes due, including three days of grace; by this practice they make the discount more than it ought to be.

The customary Rule for Discount.

Find the interest of the sum to be discounted at 5 per cent. from the day on which it is discounted to the day on which it becomes due, including 3 days beyond that date, upon a bill, and this interest will be the discount. Subtract this interest from the sum to be discounted, and the remainder will be the present worth.

Or, for each pound sterling, reckon one penny per calendar month, when the discount is at 5 per cent.

5. Thus, the discount, upon a bill of 15,000l., due 57 days after date, is 1231. 5s. 94d., being the interest of 15,000l. for (57+3 days of grace) 60 days. See Prop. 2, page 131.

6. When goods are bought or sold on which discount is to be made for present payment at any rate per cent. if no time be specified, the interest of the value of the goods for a year is the discount.

Examples.

(1.) What are the present worth and discount of 5504. 10s. for 9 months, at 5 per cent. per annum ?

16m. £5 interest of 1001. for 1 year.

2

£103 15 amount of 1001. for of a year.

103. 15s.: 1001. :: 5501. 10s.: 5301. 12s. Od. 2, the present worth; which, deducted from 550l. 10s, gives 191. 17s. 114d. for the discount.

A merchant or banker would make the discount 201. 12s. 10 d.

Or thus,

1031. 15s. 31. 15s. :: 550l. 10s.: 191. 17s. 114d., the discount; which, deducted from 550l. 10s., gives 5301. 12s. Od. for the present worth.

A merchant or banker would make the present worth 5291. 17s. 1 d. (2.) Required the present worth of 594l. 14s. 9d. due 8 months hence, allowing a discount of 52 per cent. per

annum.

(3.) Sold goods to the value of 9157. 17s. payable 7 months hence; what must I allow for present payment, at 8 per cent. per annum?

(4.) How much ready money should I have for a note of 757. which would be due 19 months hence, if I allow a discount of 5 per cent. per annum ?

CLASS II.

(5.) What is the discount of 15,000l. for 57 days, at 5 per cent. per annum?

(6.) Sold goods to the value of 8007. 16s. payable as follows, viz. at two months, at 3 months, at 9 months, at 11 months, and the rest at 12 months; what must be discounted for present payment, at 5 per cent. per annum?

EQUATION OF PAYMENTS.

Definition. When several bills are payable at different times, bearing no interest till after the term of payment, the finding a time, at which, if they are all paid together, neither the holder nor the receiver will suffer loss, is called equating, or reducing the times of payment to one.

Proposition. To find the equated time at which several bills, payable at different times, may be paid at once, without loss either to the holder or receiver, allowing simple interest.

Rule. If the times of payment be of different denominations, they must each be reduced to the same denomination. Then, multiply each payment by the time at which it becomes due; and divide the sum of the products by the sum of the payments, the quotient will be the time required.

Note 1. As this Rule of Equation of Payments has been the occasion of more disputes than all the rules of arithmetic put together, the reader will not be displeased to find here the several suppositions on which its principal defenders have founded their demonstrations.

2. Mr. Cocker supposes the equated time will be true, When the sum of the interests of the several bills which are payable before the equated time, from the times which they respectively become due to that time; is equal to the sum of the interests of the bills payable

after the equated time, from that time to the times at which they respectively become due.' But the argument by which he attempts to prove the truth of the rule is, according to Mr. Malcolm, very erro

neous.

3. Mr. Hatton supposes the equated time to be true, ‹ When the interest of the sum of the debts or bills, from the time of the question to the equated time, is equal to the sum of the interests of the several debts or bills from the time of the question to the several terms of payment; and then, by an example, shews that the rule agrees with this supposition.

4. Mr. R. Burrow, in his Diary for the year 1777, reduces the subject, To find in what time the whole sum of the single payments will produce the same amount as that which arises from the sum of all the single payments, together with the interest of each payment from the time of its becoming due to the time of the last payment; and then gives an algebraical demonstration, which shews that the rule is true according to this supposition.

5. That the rule is universally true, according to any of these suppositions, or that, if it be true according to one of them, it must necessarily be true according to the whole, may easily be demonstrated.

6. The following is KERSEY'S RULE.-Find the present worth of each debt or bill, discounting from the time at which it is payable, (by the rule of Discount,) then find (by Prop. 6. of Simple Interest) in what time the sum of these present worths will amount to the sum of the debts or bills, and that is the time sought. There are other rules given by different authors, as Sir Samuel Moreland's, Ward's, &c. ; but, upon a close attention to their principles, they will be found exactly the same as one or other of the rules already given: indeed, the foundation of Burrow's demonstration seems to have been taken from Moreland's. rule. Malcolm's rule will be given in the second part of this treatise :. its requiring an extraction of the square-root makes it inadmissible in. this place.

Examples.

(1.) A owes B 110l. whereof 50%. is to be paid at two years' end, 401. at 3 years' end, and 204. at 4 years' end; at what time may B receive the whole at once, without prejudice to either party?.

110 sum of the payments. 330 sum of the products. Then, 330 divided by 110 gives 3 years, the answer.