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3. To find the interest of 200 guineas, for 4 years, 7 months and 25 days,
4. To find the interest of 4507, for a year, at 5 per cent. per annum.
Ans. 221 10s. 5. To find the interest of 7157 12s 6d, for a year, at 4 per cent. per annum. Ans. 321 4s Old.
6. To find the interest of 7201, for 3 years, at 5 per cent. per annum.
7. To find the interest of 355/ 15s, for 4 years, at 4 per cent. per annum.
Ans. 567 18s 4 d.
8. To find the interest of 321 5s 8d, for 7 years, at 44 per cent. per annum.
9. To find the interest of
Ans. 9/ 12s 1d. for 11⁄2 year, at 5 per cent. per annum. Ans. 12/ 15s. 10. To find the insurance on 2051 15s, for of a year, at 4 per cent. per Ans. 21 1s 13d.
11. To find the interest on 3191 6d, for 52 years, at 3 per cent. per annum. Ans. 68/ 158 9žd.
12. To find the insurance on 1071, for 117 days, at 42 per cent per annum.
Ans. 17 12s 7d.
13. To find the interest of 177 5s, for 117 days, at 4 per cent. per annum.
Ans. 5s 3d.
14. To find the insurance on 7127 6s for 8 months, at 7 per cent. per annum. Ans. 35l 12s 31d.
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.
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.
RULE 1. 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. Proceed thus through all the payments to the last, always accounting the last amount as a new principal for the next payment. The reason is evident from the definition of compound interest.
RULE 2. Find the amount of 1 pound for the time of the first payment, and raise or involve it to the power of whose index is denoted by the number of payments. Then that power multiplied by the given principal, will produce the whole amount from which the said principal being subtracted, leaves the compound interest of the same. This is evident from the first rule.
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 17 for a year is or '05, and its amount 1.05. Therefore,
2. By the 2d rule.
1.05 amount of 17. 1.05
1st year's interest
2. To find the amount of 501 in 5 years, at 5 per cent. per annum, compound interest. Ans. 631 16s 34d.
3. To find the amount of 50l in 5 years, or 10 half-years, at 5 per cent. per annum, compound interest, the interest payable half-yearly. Ans. 641 Os 1d.
4. To find the amount of 50l in 5 years, or 20 quarters, at 5 per cent. per annum, compound interest, the interest payable quarterly. Ans. 647 28 0ld. 5. To find the compound interest of 370l forborn for 6 years, at 4 per cent. per annum. Ans. 981 38 44d.
6. To find the compound interest of 410l forborn for 2 years, at 4 per cent. per annum, the interest payable half-yearly. Ans. 481 48 111d.
7. To find the amount, at compound interest, of 2171 forborn for 2 years, at 5 per cent. per annum, the interest payable quarterly. Ans. 242/ 13s 44d.
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 distinguished into two cases, Alligation Medial, and Alligation Alternate.
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. 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 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.
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?
and if r denote the rate of the whole composition,
then (a+b+c) × r will be the value of the whole,
conseq. (a+b+c) × r = am + bn + cp,
and r = (am + bn + cp) ÷ (a + b + c), which is the rule.
Note. If an ounce or any other quantity of pure gold be reduced into 24 equal parts, these parts are called carats; but gold is often mixed with some base metal, which is called the alloy, and the mixture is said to be of so many carats fine, according to the proportion of pure gold contained in it: thus, if 22 carats of pure gold, and 2 of alloy be mixed together, it is said to be 22 carats 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.
2. A composition being made of 5lb of tea at 7s per lb, 9lb at 8s 6d per lh, and 141⁄2lb at 5s 10d per lb ; what is a lb of it worth? Ans 6s 10 d. 3. Mixed 4 gallons of wine at 4s 10d per gall, with 7 gallons at 5s 3d per gall, and 93 gallons at 5s 8d per gall; what is a gallon of this composition worth? Ans. 5s 4 d.
4. Having melted together 7 oz of gold of 22 carats fine, 12 oz of 21 carats fine, and 17 oz of 19 carats fine: I would know the fineness of the composition? Ans. 2019 carats fine.
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. It is, therefore, the reverse of Alligation Medial, and may be proved by it.
1. SET the rates of the simples in a column under each other. 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 compound; 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
* Demonst. By connecting the less rate with 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 other two simples managed according to the rule.
In like 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. It is obvious, from the 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 8, or any integer; 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 treble, the or part, 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 deduced that will give all the possible answers.
Thus, taking for example the four simples A, B, C, D, which are to be mixed so as to produce the mean price m. Denote the prices of A, B, C, D by m +a, m + b, m -c, and m - d respectively. Likewise let the quantities taken be x, y, z, v. Then
Then if each quantity be multiplied
(m + a) x + (m + b) y + (m − c) ≈ + (m − d) v = m (x + y + z + v).
That is, axby=cz + dv.
But as there are four unknown quantities, and but one equation, we are at liberty to assume any other three conditions we please, and still the true result will be obtained. The three simplest that can be taken are those upon which the rule above given is founded, viz.
x=d, y = c, z = b, v = a; or x = c, y=d, z = a, v = b.
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.
1. A merchant would mix wines at 16s, at 18s, and at 22s per gallon, so that the mixture may be worth 20s the gallon; what quantity of each must be taken?
2. How much sugar at 4d, at Ed, and at 11d per lb, must be mixed together, so that the composition formed by them may be worth 7d per lb?
Ans. 1 lb. or 1 stone, or 1 cwt, or any other equal quantity of each sort. 3. How much corn at 2s 6d, 3s 8d, 4s, and 4s 8d per bushel must be mixed together, that the compound may be worth 3s 10d per bushel?
Ans. 2 at 2s 6d, 3 at 3s 8d, 3 at 4s, and 3 at 4s 8d.
WHEN the whole composition is limited to a certain quantity: find an answer as before by linking; then say, as the sum of the quantities, or differences thus determined, is to the given quantity; so is each ingredient, found by linking, to the required quantity of each.
1. How much gold of 15, 17, 18, and 22 carats fine, must be mixed together, to form a composition of 40 oz of 20 carats fine.
Ans. 5 oz of 15, of 17, and of 18 carats fine, and 25 oz of 22 carats fine *.
which is the rule, and any other three relations would give a different but a less practicable rule than this. The next in point of simplicity is
x = pd, y = qc, z = : qb, v = pa; or, x = pc, y = qd, z = pa, v = qb.
See the Key to Keith's Arithmetic, by Maynard, from which valuable little treatise the latter part of this note is taken.
* A great number of questions might be here given relating to the specific gravities of bodies, but one of the most curious may suffice.
Hiero, king of Syracuse, gave orders for a crown to be made entirely of pure gold; but,