7. To find the interest of 355 15s for 4 years, at 4 per cent per annum Ans. 56/18, 43d. E. 8 To find the interest of 321 5s 8d, for 7 years, at 4 per cent. per annum.

9. To find the interest of 1701, for 1 year,

per annum.

10 To find the insurance on 2051 158, for 4 per cent. per annum.

Ans. 9 128 1d. at 5 per cent.

Ans. 12/ 58.

of a year, at Ans. 2/18 13d.

11. To find the interest of 3191 6d, for 5 years, at 3 per cent per annum. Ans. 68/ 58 9jd. 12. To find the insurance on 2071, for 117 days, at 44 per cent per annum. Ans 1/12.7d. 13 To find the interest of 171 58, for 117 days, at 43 per cent. per annum. Ans. 5s 3d. 14. To find the insurance on 712/ 68, for 8 months, at 73 per cent. per annum. Ans 35 123 34d. Note. The Rules for Simple Interest, serve also to calculate Insurances, or the Purchase of Stocks, or any thing else that is rated at so much per cent.

See also more on the subject of Interest, with the algebraical expression and investigation of the rules at the end of the Algebra, next following.

COMPOUND INTEREST.

COMPOUND INTEREST, called also Interest upon Interest, is that which arises from the principal and interest, taken together, as it becomes due, at the end of each stated time of payment. Though it be not lawful to lend money at Compound Interest, yet in purchasing annuities, pensions, or leases in reversion, it is usual to allow Compound Interest to the purchaser for his ready money.

RULES Find the amount of the given principal, for the time of the first payment, by Simple Interest. Then consider this amount as a new principal for the second payment, whose amount calculate as before. And so on through all the payments to the last, always accounting the last amount as a new principal for the next payment. The reason of which is evident from the definition of Compound Interest. Or else,

2. Find the amount of 1 pound for the time of the first payment, and raise or involve it to the power whose index is denoted by the number of payments. Then that power multiplied by the given principal, will produce the whole

amount.

amount. From which the said principal being subtracted, leaves the Compound Interest of the same. As is evident from the first Rule.

EXAMPLES.

1. To find the amount of 7201, for 4 years, at 5 per cent. per annum,

Here 5 is the 20th part of 100, and the interest of 1 for a year is or05, and its amount 1:05. Therefore,

20

1. By the 1st Rule.

2. By the 2d Rule.

1.05 amount of 17. 1.05

1-1025 2d power of it.

1.025

1 21550625 4th pow. of it. 720

3d yr's princip. 39 13 9 3d yr's interest.

2. To find the amount of 50%, in 5 years, at 5 per cent. per annum, compound interest Ans. 63 168 34d.

3. To find the amount of 501 in 5 years, or 10 half-years, at 5 per cent per annum, compound interest, the interest payable half-yearly Ans. 64/ Os 1d.

4. To find the amount of 501, in 5 years, or 20 quarters, at 5 per cent. per annum, compound interest, the interest payable quarterly. Ans. 641 28 0ld. 5. To find the compound interest of 370 forborn for 6 years, at 4 per cent. per annum. Ans. 98/ 38 4jd

6. To find the compound interest of 410/ forborn for 23 years, at 41⁄2 per cent. per annum, the interest payable halfyearly. Ans. 48/ 48 114d. 7. To find the amount, at compound interest, of 2171, forborn for 24 years, at 5 per cent per annum, the interest payable quarterly. Ans. 242/ 138 41d. Note. See the Rules for Compound Interest algebraically investigated, at the end of the Algebra,

ALLIGATION.

ALLIGATION.

ALLIGATION teaches how to compound or mix together several simples of different qualities, so that the composition may be of some intermediate quality or rate. It is commonly distinguish into two cases, Alligation Medial, and Alligation Alternate.

ALLIGATION MEDIAL.

ALLIGATION MEDIAL is the method of finding the rate or quality of the composition, from having the quantities and rates or qualities of the several simples given. And it is thus performed:

* MULTIPLY the quantity of each ingredient by its rate or quality; then add all the products together, and add also all

* Demonstration. The rule is thus proved by Algebra.

Let a, b, c be the quantities of the ingredients, and m, n, p their rates, or qualities, or prices;

then am, bn, cp are their several values,

and am + bn + cp the sum of their values,

also a+b+c is the sum of the quantities,

and if denote the rate of the whole composition,

then a+b+cXr will be the value of the whole,

conseq. a + b + c × r = am + bn + cp,

andram + bn + cp ÷ a+b+c, which is the Rule.

Note, If an ounce or any other quantity of pure gold be reduced in to 24 equal parts, these parts are called Caracts; but gold is often mixed with some base metal, which is called the Alloy, and the mixture is said to be of so many caracts fine, according to the proportion of pure gold contained in it; thus, if 22 caracts of pure gold, and 2 of alloy be mixed together, it is said to be 22 caracts fine

If any one of the simples be of little or no value with respect to the rest, its rate is supposed to be nothing; as water mixed with wine, and alloy with gold and silver.

the quantities together into another sum; then divide the former sum by the latter, that is, the sum of the products by the sum of the quantities, and the quotient will be the rate or quality of the composition required.

EXAMPLES.

1. If three sorts of gunpowder be mixed together, viz. 50lb at 12d a pound. 44lb at 9d, and 26lb at 8d a pound; how much a pound is the composition worth?

Here 50, 44, 26 are the quantities,

and 12, 9, 8 the rates or qualities;
then 50 X 12 = 600

44 X 9396

26 X 8=208

Ans. The rate or price is 10d the pound.

2. A composition being made of 5lb of tea at 78 per lb, 9lb at 88 6d per lb, and 144lb at 5s 10d per lb; what is a Ib of it worth? Ans. 68 10d.

3. Mixed 4 gallons of wine at 48 10d per gall, with 7 gallons at 58 3d per gall, and 9 gallons at 58 8d per gall; what is a gallon of this composition worth? Ans. 58 41d.

4. A mealman would mix 3 bushels of flour at 38 5d per bushel, 4 bushels at 58 6d per bushel, and 5 bushels at 48 8d per bushel; what is the worth of a bushel of this mixture? Ans. 48 74d.

5. A farmer mixes 10 bushels of wheat at 58 the bushel, with 18 bushels of rye at 3s the bushel, and 20 bushels of barley at 2s per bushel : how much is a bushel of the mixture worth? Ans. 3s.

6. Having melted together 7 oz of gold of 22 caracts fine, 12 oz of 21 caracts fine, and 17 oz of 19 caracts fine: I would know the fineness of the composition?

Ans. 20 caracts fine.

7. Of what fineness is that composition, which is made by ixing 3lb of silver of 9 oz fine, with 5lb 8 oz of 10 oz fine, and 11b 10 oz of alloy?

Ans. 783 oz fine.

ALLIGATION

ALLIGATION ALTERNATE.

ALLIGATION ALTERNATE is the method of finding what quantity of any number of simples, whose rates are given, will compose a mixture of a given rate. So that it is the re verse of Alligation Medial, and may be proved by it.

RULE IP.

1. SET the rates of the simples in a column under each ether -2. Connect, or link with a continued line, the rate of each simple, which is less than that of the compound, with one, or any number, of those that are greater than the com pound; and each greater rate with one or any number of the less.-3 Write the difference between the mixture rate, and that of each of the simples, opposite the rate with which they are linked-4 Then if only one difference stand against any rate, it will be the quantity belonging to that rate; but if there be several, their sum will be the quantity.

The examples may be proved by the rule for Alligation Medial.

* Demonst. By connecting the less rate to the greater, and placing the difference between them and the rate alternately, the quantities resulting are such, that there is precisely as much gained by one quantity as is lost by the other, and therefore the gain and loss upon the whole is equal, and is exactly the proposed rate and the same will be true of any o her two simples managed according to the Rule,

In tike manner, whatever the number of simples may be, and with how many soever every one is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss and gain between every two, and consequently an equal balance on the whole. Q. E. D.

It is obvious, from this Rule, that questions of this sort admit of a great variety of answers; for, having found one answer, we may find as many more as we please, by only multiplying or dividing each of the quantities found, by 2, or 3, or 4, &c: the reason of which is evident; for, if two quantities, of two simples, make a balance of loss and gain, with respect to the mean price, so must also the double or trebie, the orpart, or any other ratio of these quantities, and so on ad infinitum.

These kinds of questions are called by algebraists indeterminate or unlimited problems; and by an analytical process, theorems may be raised that will give all the possible answers.

XAMPLES