Here, the number of days being 134, we have this analogy: as 365 days: 134 days:: £5: £1-16-81, the interest of £100 for 134 days; and then, as £101 - 16 - 8: £100:: £512: £502 - 15 - 51,

answer.

This question, and all others in which it is required to find the true present worth of a sum for a given number of days, may be wrought more easily and more accurately by the following rule:

RULE IV. Multiply the days by the rate, and add the product to 36,500 (= 365 × 100); then, as this sum is to 36,500, so is the debt to its true present worth.

Thus, in the present example, 134 × 5670, and 36,500 +670 =37,170: then as 37,170: 36,500::£512: £502 - 15 - 5, the present worth.

365

This rule depends on the same principle as the last; and the reason of it may be thus illustrated. As 365 days: 134 days:: £5 £87, the interest of £100 for 134 days, at 5 per cent. per annum. Then, as £100879: £100; or by reduction to three-hundred-andsixty-fifths, as 37,160 is to 36,500, so is the debt to its present worth. Both these rules are, in reality, the same as Rule VI. in Interest.*

365

The following considerations will be useful in showing the falsity of the common method of discount.

If a person have a bill for £100, payable at the end of a year, at 5 per cent., he will receive, according to the common method of discount, only £95 for it; and were he to lend this sum for a year, at the same rate, instead of £100, to which it obviously should amount, he would receive only £99 - 15. The true present worth is £954-9, and, consequently, the error 4s. 94d. Again, had the same bill been payable at the end of two years, the present worth, by the common method, would have been £90, while it should be £90-18-22. The error is consequently 18s. 22d., and £90, instead of amounting to £100 at the end of two years, would amount to no more than £99. Had the time been four years, the present worths would have been £80, and £83 - 6-8, and the error £3-6-8. The amount also of the present worth, £80, would be £96, and, consequently, £4 less than it should be. If the time had been 10 years,

* Unless the time be great, the true present worth may be readily derived by approximation from that found by the common method, by finding the interest of the interest first found, and adding it to the present worth found by the common method; then, by finding the interest of this last interest, and subtracting it from the approximate present worth; and so on, by adding and subtracting alternately the interest of the last interest, till the correction becomes so small, that it would be unnecessary to carry the operation farther. When the time is very short, the true result will often be obtained as easily in this way, as by the principles above explained.

As an example, let it be required to find the present worth of a bill of £140, due at the end of 6 months, at 4 per cent. per annum.

Here the interest of £140 is £2 - 16; that of £2 - 16 is 1s. 1d.; and that of 1s. 1d. is a farthing. Then, by subtracting £2-16 from £140; by adding 1s. 1 to the remainder; and, lastly, by subtracting a farthing from that result, we find the present worth to be £137-5-14, which is correct.

the present worths would have been £50, and £66 - 13 - 4, where the error is £16-13-4; and the amount of the present worth, £50, would be £75, instead of £100. Finally, were the time 20 years, the present worth, according to the common method, would be nothing, while it should be £50; and were the time greater than 20 years, the present worth would be unassignable, as it would appear to be less than nothing; or, if any meaning could be attached to the result of the operation, it would be, that the person who held the bill, instead of receiving anything for it, would be required to pay something to get it off his hands.

From these examples, it will appear how very erroneous the common method of computing discount is, when the time is long.* In every case the discounter of the bill has a greater rate of interest for his money than the nominal one; and the longer the time, the greater is this rate. Thus, to recur to the last series of examples, since, by paying £95 at present, the discounter will be entitled to £100 at the end of a year, he obviously gains £5 on £95; and, therefore, £95: £5:: £100 : £5 - 5 - 3, his gain per cent. In like manner, if the time were two years, the gain per cent. would be found to be £5 - 11 - 1; if four years, £65; if 10 years, 10 per cent.; and if 19 years, cent. per cent.

It is true, indeed, that when the time is short, as it generally is in real business, the results found by the two methods are pretty nearly the same; and, therefore, the common method, the computation for which is so easy, may be employed without much error. Still, however, the principle is false, as it gives profits to the discounter which are not proportional to the times. It may be said, that those who keep money for the purpose of discounting are entitled to more than the simple common rate. This may be true; but, if the discounter is to have a greater rate, it should be a fixed one, not depending on the time the bill has to run.

Should the learner wish to work discount in the correct method, the exercises at the beginning of this article will serve his purpose as well as any others; and the following are their answers by that method :

*It might be shown algebraically, that the error in the common method of calculating the discount or present worth of a given debt, at a given rate, is nearly proportional to the square of the time, when the time is small, or, more properly, when the discount is small compared with the debt. Thus, at 5 per cent. per annum, the error on a bill of £1000, for 2 months, is nearly 1s. 41d., while, for 4 months, it is nearly 5s. 51d., or very nearly 4 times 1s. 41d.

It might also be shown, that the error in the sums to which the present worth of a given sum, found by the common method, would amount at the given rate, would be exactly proportional to the square of the times. Thus, in the examples in the text, in last page, it appeared that in case of a bill of £100, payable in a year, at 5 per cent. per annum, the amount would be £99 - 15, while, if it were payable in two years, the amount would be only £99, the error being in the one case 5s., and in the other £1, or 4 times 5s.

COMMISSION, INSURANCE, &c.

COMMISSION is the sum which a merchant charges for buying or selling goods for another.

BROKERAGE is a smaller allowance of the same nature, paid usually for negotiating bills, or transacting other money concerns.

INSURANCE, or ASSURANCE, is a contract by which one party, on being paid a certain sum, or PREMIUM, by another, on account of property that is exposed to risk, engages, in case of loss, to pay to the owner of the property the amount of loss, if it do not exceed the sum insured on the property.

RULE I. To compute the commission, brokerage, insurance, or any other allowance on a given sum, at a given rate per cent.: Multiply the sum by the rate per cent., and divide the product by 100; or, as £100 is to the rate per cent., so is the given sum to the required allowance.

The work is performed as in the case of simple interest for a year at a given rate per cent.

RULE II. To find how much must be insured on property worth a given sum, so that, in case of loss, both the value of the property and the premium may be repaid: (1.) Subtract the rate from £100. (2.) As the remainder is to £100, so is the value of the property to the sum to be insured.

The work is performed by the rules of Simple Proportion.

M

PROFIT AND LOSS.

That branch of arithmetic which treats of the gains or losses on mercantile transactions, is called profit and

LOSS.

The work is performed by Simple Proportions, stated according to the following rules :

.

RULE I. From the prime cost and the selling price, to find the gain or loss per cent. As the prime cost is to the gain or loss on that cost, so is £100 to the gain or loss per cent.

RULE II. To find how a commodity must be sold to gain or lose a certain rate per cent.: As £100 is to the gain or loss per cent., so is the prime cost to the gain or loss on that cost; and from this and the prime cost, the selling price will be found by addition or subtraction.

It should be particularly remarked, that, by the gain or loss per cent. is to be understood the sum that would be gained or lost at the given prices, not on a hundred pounds' worth sold, but on a hundred pounds laid out in prime cost, and in charges, if there be any.

RULE III. From the gain per cent., and the selling price, to find the first cost: As £100, together with the gain per cent., or diminished by the loss per cent., is to £100, so is the selling price to the prime cost.

EXCHANGE.

The object of EXCHANGE is to find how much of the money of one country is equivalent to a given sum of the money of another.

This amount depends partly on the mint regulations in the two countries, and partly on the course of trade. So far as the exchange depends on the mint regulations, it generally continues the same for a number of years, whilst so far as it depends on the course of trade, it fluctuates from day to day. The first, or comparatively constant part of the exchange, is called the PAR OF

EXCHANGE, whilst the whole ever-varying amount is called the cOURSE OF EXCHANGE.

The PAR OF EXCHANGE between two countries is the number of standard coins of one country which is equal in value to one of the standard coins of the other country, if the values of gold and silver are in an assumed constant proportion, and if the value of each metal be the same in both countries.

In some countries gold coins are the standard of value, as in British currency since 1816, in Germany and the Scandinavian States since 1870, while in some, as in British India, silver coins are the standard. The weight of these coins is fixed by the mint regulations of the respective countries, as also their fineness, or proportion of pure gold or silver to alloy in them. Between two countries which both have a gold standard, as Great Britain and Germany, on the assumption that gold is of the same value in both countries, it is easy, by compound proportion, to ascertain how much of the standard coin of account of one country is equal to one of the standard of the coin of account of the other country. As, for instance, how many marks (reichsmarks) and pfennige are equal to £1 or British sovereign? The amount so ascertained is the par of exchange between the countries.

If the par of exchange has to be calculated between two countries where one has a gold standard, like Great Britain, and the other, like India, has a silver standard, it is necessary to take into account, besides the weight and fineness of the respective coins which form the standard of account, the proportion between the values of equal weights of gold and silver. This proportion, though varying considerably after the continuous discovery of very fertile mines, and varying slightly from day to day, is for long periods usually very constant. In the mints of various countries some proportion is generally assumed as a guide for the mint arrangements. The published par of exchange is sometimes calculated at the proportion so assumed, and sometimes at the current price of silver in the market where a gold standard is used. The information as to mint regulations necessary to calculate the par of exchange is published in books on coins and exchanges called Cambists. As the same books usually give the par of exchange for some stated proportions of the precious metals, merchants may do a great deal of business with foreign countries without having to perform the calculation of the par of exchange.

If the value of either of the precious metals should undergo a temporary change, so as to alter the proportion on which the usual par is calculated, it is easy for bullion dealers and others interested to compute the par at the altered proportion, for the new par is to the par in books in the exact proportion of the observed ratio between the values of equal weights of gold and silver to the ratio assumed in the par in books. For the more advanced student the method of calculating the par of exchange is given immediately after the