us if it were the right number: then say, As the result of this work is to the position, so is the result in the question to the number required.

Ex. 1. A person counting some guineas, being asked how many he had, replied: "If you had as many, and as many more, and half as many, and one quarter as many, you would have 264." How many had the person who was counting his gold?

By way of supposition, I take 80 as the number; then, by the terms of the question, it will be

Ex. 2. A person, after spending 1, 1, and 4th of his money, finds he had 5007. left, what was his original property?

I take a number divisible by 2, 4, and 6, for the supposition,

viz. 60.

Ex. 3. Three persons bought goods at Manchester which cost 600l. The first person was to have a third part more than the second, and the third a fourth part more than the first; what was each man's share?

Ex. 4. In a leaky vessel there were three pumps of different capacities; the first would empty the hold of the ship in 20 minutes, the second would require double that time, and the third would not perform the business in less than an hour; how long would all three together take in doing it?

NOTE.

* Any other number, as 12 for instance, would have answered the same purpose: then it would have been 12 - 11 = 1, and

1: 12: 500: 6000l.

DOUBLE POSITION.

QUESTIONS in this rule are resolved by making suppositions of two numbers, which may both prove false; in that case the errors are made to correct each other.

RULE. (1) Place each error against its respective position, and multiply them cross ways. (2) If the errors are alike, that is, both greater or both less than the given number, take their difference for a divisor, and the difference of their products for a dividend. But if unlike, take their sum for a divisor, and the sum of their products for a dividend, the quotient will be the answer.

Ex. 1. Three persons have obtained the 20,0001. prize in the lottery, and it is to be so divided, that the second is to have 6007. more than the first, and the third 8007. more than the second, what is each person's share? Suppose the first had 5000 Then the second had 5600

Suppose the first had 5600

The second had
The third had

6200

and the third had 6400

7000.

Ex. 2. A gentleman, at Christmas, wished to give several poor fa❤ milies 5 shillings each, but he found he had 163. 8d. too little; he then gave them 3s. 6d. each, and found he had 4s. 4d. left, how many families were there?

Ex. 3. A person purchased a house and land, together with a carriage and horses, for 1500l. ; he paid 4 times the price of the carriage and horses for the land, and 5 times the price of the land for the house, what was the value of each separately?

COMPOUND INTEREST AND ANNUITIES.

COMPOUND INTEREST, or interest upon interest, is that which is paid not only for the use of the money lent, but also for the use of the interest as it becomes due.*

There are three methods of working Problems in this Rule, viz. by Common Arithmetic; by Decimals; and by Logarithms: I shall give examples under each.

I. By Common Arithmetic.

RULE 1. Find the amount of the given principal for the time of the first payment by simple interest. 2. Consider this amount as the principal for the second payment, the amount of which is to be calculated as before, and so on through all the payments to the last, still reckoning the last amount as the principal for the next payment.

Ex. 1. What is the amount of 5501. for three years, at 5 per cent. compound interest ?

Ex. 2. What is the amount of 400l. for four year's, at 5 per cent. compound interest?

Ex. 3. What is the compound interest of 600l. for five years, at 5 per cent. compound interest ?+

NOTES.

*It is not lawful to lend money at compound interest: but in granting or purchasing annuities, leases, or reversions, compound interest for money is allowed.

+ Here, when the amount is found, the principal must be taken from it, and the remainder is the compound interest. Thus, in the first example, the compound interest is 6361. 13s. 10žd. — 550l., or 96l. 13s. 10 d. In short periods compound interest differs but little from simple interest; in this case, for instance, the simple interest

II. By Decimals.

RULE 1. Find the amount of 11. for a year, at the given rate per cent. 2. Involve the amount thus found, to such a power as is denoted by the number of years.. 3. Multiply this power by the principal, or given sum, and the product will be the amount required. 4. Subtract the principal from the amount, and the remainder will be the interest. Ex. 1. What is the compound interest of 5501. for 3 years, at 5 per cent. per annum ?

1.05 amount of 11. for a year, at 5 per cent.; Then 1.05 X 1.05 X 1.05 1.157625, and

1.157625 X 550 636.69375 amount, 636.69375 55086.69375 — 867. 13s. 101d.

Ex. 2. What is the amount of 400l. for 4 years, at 5 per cent. per annum ?

Ex. 3. What is the compound interest of 620l. for 5 years, at 5 per cent.?

III. By Logarithms.

RULE. Multiply the logarithm of the amount of 11. for a year, by the number of years, and to the product add the logarithm of the principal, and the answer is the amount required.

Ex. 1. What is the amount of 5501. for 3 years, at 5 per cent. per ann. compound interest?

0.0211893 log, of 1.05

would be 82l. 10s. When, however, the interest is suffered to accumulate for many ages, the difference between simple and compound interest is almost beyond belief. To give an example, we have seen, p. 164, that a penny put out to simple interest at the birth of Christ would, at the end of the year 1800, only amount to about 7s. 6d.; whereas the same sum, for the same period, at Compound Interest, would have amounted to a sum greater than could be contained in six hundred millions of globes, each equal to the carth in magnitude, and all of solid gold.

* See p. 142.

The nearest logarithm to 2.8039306 is 2.8034571, which answers to 636; the difference between the logarithms is 4735, which I

Ex. 2. What is the amount of 400l. for four years, at 5 per cent. per annum compound interest?*

Ex. 3. What is the compound interest of 6091. for 5 years, at 5 per cent. per annum?

Ex. 4. What is the amount of 8451. for 14 years, at 5 per cent. compound interest?

We shall now proceed to consider this subject more generally by the help of tables, the construction of which will be shewn in the note below.

NOTE.

multiply by 20 to bring into shillings, and divide by the common difference found at the margin of the table; thus,

4735
20

6823)94700(13

88699

6001

12

6823)72012(10

68230

3782
4

6823)15126

13646

1482

We have purposely given the same examples in all three methods, in order that the pupil may compare the difference in working.

Although we have intentionally abstained from the use of symbols or letters, yet it seems necessary to give some account of the tables by which Compound Interest, Life Annuities, Reversions, &c. are calculated. This will be perfectly intelligible to those who have made themselves masters of what is gone before; and in a work on Algebra, preparing for the press, will be given a demonstration of the rules of common arithmetic :

Let r the amount of 11. for one year,

n the number of years,

p

principal,

aamount, or principal and interest of the given sum for the time required.

By proportion we say, as 11. principal is to the amount of 17. for one year, so is the amount of 17. for one year, to the amount for two years, and so on: thus

1 : 7 :: : ? 17 :: 2 : ༡་9 1:r:: 73 : gos 1: 7: 71

amount or 1. in 2 years.
amount of 17 in 3 years.
amount of 14. in 4 years.
amount of 17. in 5 years.