ILLUSTRATION.

Suppose the interest of money to be at 5 per cent. and that 3 years is the true equated time as found above. It is evident that A gains the interest of 501. for one year, which is 21. 10s., by extending the term of payment to 3 years instead of 2; and that he loses the interest of 401. for half a year, and the interest of 201. for 1 year, by paying 401. half a year before it comes due, and 201. 1 year before it becomes due; which interests, added together, make 21. 10s., so that his gain and his loss, on this consideration, appear to be equal. But, we must recollect, that B-is not intitled to the interest of 401. for half a year, and of 201. for 1 year, but to the discount of each of these sums for those times; so that the rule cannot be precisely accurate, though it be near enough to the truth for any practical purpose to which it can be applied.

(2.) I am to pay 5007. at three different payments, viz. 1007. at 2 months, 2007. at 4 months, and the rest at 6 months; but the person who is to receive the money has agreed to take a single note for the payment of the whole at once, for what length of time must the note be given?

(3.) A debt of 7007. is to be discharged thus: 1507. present, 300l. at 6 months, 2001. at 9 months, and the rest at 12 months; what is the equated time for the payment of the whole?

(4.) A merchant buys goods to the amount of 7501. 3501. of which is to be paid at 3 months, and the rest at 9 months; to prevent farther trouble, it is agreed to pay the whole at once, and to prolong the time of the first payment in proportion to the shortening the time of the second; at what time must the whole be discharged without prejudice to either?

(5.) A debt of 500l. 15s. is payable as follows: 1507. at 2 months, 1477. 17s. at 74 days, 1374. 18s, at 95 days, and the rest at 5 months. It is to be discharged at one payment; what is the equated time, reckoning 30 days to a month?

CLASS II.

(6.) A traveller received 1200 in 4 bills, all payable at Newcastle-upon-Tyne; viz. 6007. due at 4 months, 3001. at 5 months, 2007. at 7 months, and 1007, at 10

months: he agreed to pay the banker there, a reasonable commission, and the expense of the stamps, provided he would give him a single bill on London for the payment of the whole at once; for what length of time after date ought this bill to be drawn?

(7.) A debt is to be discharged thus, present, at 25 days, at 3 months, and the rest at 4m. 17d. what time may the whole be paid at once?

(8.) Three legacies are left by a gentleman, in his will, payable by his executors, to one person, or his heirs. The first legacy of 500l. 18s. is payable in a year, the 2d of 900l. 17s. 6d. is payable in 1 year 114 days, and the 3d of 17007. 18s. 4ąd. is payable in 2 years. The legatee and executors have agreed, that the payment of these sums shall be made at once; at what time must that be, that neither party may be injured, allowing simple interest?

COMPOUND INTEREST.

Definition.-Compound Interest is that which is produced not only from the sum of money lent as the principal, but also from the interest, which, (when unpaid,) as it becomes due, is added to the principal.

Proposition. To find the interest of any sum of money, unpaid, for any equal number of payments at any rate per cent.

Rule I. Find the amount of the given principal for the time of the first payment by Simple Interest; then consider this amount, as the principal for the second payment, and find its amount as before. Proceed thus through all the payments, always considering the last amount as the principal of the next payment; then, if the given principal, or money lent, be deducted from the last amount, the remainder will be the interest required.

Or, Rule II.

Reduce the given sum into farthings, which multiply by the rate per cent. and cut off two figures from the right hand of each successive product, (or place each successive product two figures farther towards the righthand,) and the last result will be farthings.

Note. The above rules will be true, whether the payments are made yearly, half-yearly, quarterly, monthly, or by any other aliquot part of a year: thus, for half-yearly payments, take half the rate per cent., and twice the number of years ;--for quarterly payments, take of the rate per cent., and four times the number of years, &c. But the given time must be complete years, half-years, or quarters; thus, you cannot find the interest of a given sum payable yearly, for 41 years, 4 years, &c. by the above rules, as directed by several authors. The truth of this remark will easily appear to those who are acquainted with logarithmical arithmetic, and the involution of numbers to fractional powers.-For other rules, see Compound Interest. by Decimals,

Examples.

(1.) What is the compound interest of 3577. 10s. for 3 years, at 5 per cent. per annum?

more than the simple interest of the same sum. See Example 1, Simple Interest.

(2.) What is the compound interest of 700l. 188, for 4 years, at 5 per cent. per annum?

(3.) What is the compound interest of 10577. 17s. 6d. for 6 years, at 4 per cent. per annum?

(4.) Required the amount of 500l. 17s. for 5 years, at 4 per cent. compound interest?

(5.) What will 700l. amount to in 7 years, at 43 per cent. per annum, compound interest?

CLASS II.

(6.) Find the several amounts of 5007. payable yearly, half-yearly, and quarterly, for 4 years, at 5 per cent. per annum. Answ. 6077. 15s. Ožd. for yearly, 6091. 4s. 04d. for half-yearly, and 6091. 18s. 10 d. for quarterly pay

ments.

(7.) What is the amount of 7157. for 6 years, the interest payable half-yearly, at 44 per cent. per annum? (8.) What is the compound interest of 740l. 18s. for 9 years, by quarterly payments, at 4 per cent. per

annum?

FELLOWSHIP, OR PARTNERSHIP.

Definition. Fellowship, or Partnership, is a general rule by which the accounts of merchants, &c. trading in company, with a joint stock, are adjusted; so that every partner may have his due share of the gain, or sustain a proportional part of the loss, according to the money he' has advanced in the stock, and the time of its continuance therein.

SINGLE FELLOWSHIP, OR PARTNERSHIP FOR ANY EQUAL TIME.

Definition. Single Fellowship, or Partnership for any equal time, is when different stocks are employed for any certain equal time. The effects of bankrupts are by this rule properly divided among their creditors, legacies adjusted in deficiencies of assets, &c.—It likewise teaches us to divide any given number into unequal parts, proportional to certain other given numbers.

Proposition. Having each man's particular stock and the whole gain or loss given, to find each man's part of the gain or loss.

Rule. As the whole stock is to the whole gain or loss, so is each man's particular stock to his particular share of the gain or loss.

Method of proof. Add all the shares together, and the sum will be equal to the given gain or loss when the work is right.

Note 1. When there are many partners concerned, the following rule, which is best performed by decimals, will be found useful. Divide the whole gain, or loss, by the whole stock, and the quotient will be a common multiplier, by which multiply every man's parti cular stock, and the several products will give each man's share of the gain or loss.

2. Proposition. To divide any given number into any number of unequal parts proportional to certain other given numbers.

Rule. Make the sum of the numbers to which the required parts must be proportional, the first term; the number to be parted, or divided, the second; and each of the given numbers, to which the required ones must be proportional, the several third terms of so many statings in the Rule of Three, the fourth terms of which will be the respective parts required.

Examples.

(1.) Three merchants, A, B, and C, enter upon a joint adventure; A puts into the common stock 250l. 10s. B, 300l. 15s. and C. 410l. 18s. After all expences were paid, a clear gain of 3271. 11s. 6d. was to be divided amongst them; what was each man's share?

962 3: 327 11 6 :: 410 18: 139 17 10-15239, C's share.

327 11 6 proof.

(2.) Two merchants traded together; A pat into the stock 5007. 17s. 10d. and B 700 guineas; they gained 3007. 15s. what is each person's share thereof?