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.

Although 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.—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. 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 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. 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 14. for a year, is or 05, and its amount 105. Therefore,

2. To find the amount of 50%., in 5 years, at 5 per cent. per annum, compound interest.

Ans. 631. 16s. 3 d.

at 5 per cent. per Ans. 647. Os. 1d. at 5 per cent. per Ans. 641. 2s. 04d. years, at 4 per cent. Ans. 981. 3s. 41α

3. To find the amount of 50%, in 5 years, or 10 half years, annum, compound interest, the interest payable half yearly. 4. To find the amount of 50%, in 5 years, or 20 quarters, annum, compound interest, the interest payable quarterly. 5. To find the compound interest of 370%, forborn for 6 per annum. 6. To find the compound interest of 4104, forborn for 2 years, at 41⁄2 per cent. per annum, the interest payable half yearly.

7. To find the amount, at compound interest, of 2177., at 5 per cent, per annum, the interest payable quarterly.

Ans. 481. 4s. 114d. forborn for 24 years, Ans. 2424 138. 4.d

POSITION.

POSITION is a method of performing certain questions, which cannot be resolved by the common direct rules. It is sometimes called False Position, or False Supposition, because it makes a supposition of False numbers, to work with, the same as if they were the true ones, and by their means discovers the true numbers sought. It is sometimes also called Trial and Error, because it proceeds by trials of false numbers, and thence finds out the true ones by a comparison of the errors.

I osition is either Single or Double.

SINGLE POSITION.

SINGLE POSITION is that by which a question is resolved by means of one supposition only.

Questions which have their results proportional to their suppositions, belong to Single Position; such as those which require the multiplication or division of the number sought by any proposed number; or when it is to be increased or diminished by itself, or any parts of itself, a certain proposed number of times.

RULE.-Take or assume any number for that required, and perform the same operations with it, as are described or performed in the question.

Then say, as the result of the said operation, is to the position, or number assumed; so is the result in the question, to the number sought.*

EXAMPLES.

1. A person, after spending and of his money, has yet remaining 604; what had he at first?

Then, 50 120 :: 60: 144, the answer.

2. What number is that, which multiplied by 7, and the product divided by 6, the quotient may be 14?

Ans. 12

* The reason of the rule is evident, because it is supposed that the results are proportional to the suppositions.

3. What number is that, which being increased by 4, 4, and of itself, the sum shall be 125?

4. A general, after sending out a foraging, and § of his men, had ing 700; what number had he in command ?

Ans. 60.

yet remainAns. 4200.

5. A gentleman distributed 78 pence among a number of poor people, consisting of men, women, and children; to each man he gave 6d,, to each woman 4d., and to each child 2d.: moreover there were twice as many women as men, and thrice as many children as women. How many were there of each? Ans. 3 men, 6 women, and 18 children. 6. One being asked his age, said, if of the years I have lived, be multiplied by 7, and of them be added to the product, the sum will be 292. What was his age? Ans. 60 years.

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DOUBLE POSITION.

DOUBLE POSITION is the method of resolving certain questions by means of two suppositions of false numbers.

To the Double Rule of Position belong such questions as have their results not proportional to their positions: such are those, in which the numbers sought, or their parts, or their multiples, are increased or diminished by some given absolute number, which is no known part of the number sought.

RULE 1.—Take or assume any two convenient numbers, and proceed with each of them separately, according to the conditions of the question, as in Single Position; and find how much each result is different from the result mentioned in the question, noting also whether the results are too great or too little.

Then multiply each of the said errors by the contrary supposition, namely, the first position by the second error, and the second position by the first error. If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer.

But if the errors are unlike, divide the sum of the products by the sum of the errors, for the answer.

Note. The errors are said to be alike when they are either both too great, or both too liule; and unlike, when one is too great and the other too little.

• Demonstration.-The rule is founded on this supposition, namely, that the first error is to the second, as the difference between the true and first supposed number, is to the difference between the true and second supposed number; when that is not the case, the exact answer to the question cannot be found by this rule.-That the rule is true, according to that supposition, may be thus proved.

Let a and b be the two suppositions, and A and B their results, produced by similar operations; also r and a their errors, or the differences between the results A and B from the true result N; and iet denote the number sought, answering to the true result N of the question.

Then, is NA=r, and NB. And, according to the supposition on which the rule is founded, r::: -a: z-b; hence, by multiplying extremes and means, rx — ro = 83 — 8α ; then, by transposition, rx — sx = rb — sa; and, by division, ☛ = which is the rule when the results are both too little.

rb-a

-

the number sought,

If the results be both too great, so that A and B are both greater than N; then NA = −r, and N-B-, or and s are both negative; hencer : - 8 : : 2 - 0 : ≈ — b, but — r : — 8 : +:+1, therefore r ; ; : : x − a : x-b, and the rest will be exactly as in the former case. But if one result A only be too little, and the other B be too great, or one error r positive, and the others negative, then the theorem becomes a = which is the rule in this case, or when the errors are unlike

r+s

EXAMPLE,

1. What number is that, which being multiplied by 6, the product increased by 18, and the sum divided by 9, the quotient shall be 20.

Suppose the two numbers, 18 and 30. Then,

First position,

Sum

8) 216 sum of products.

RULE II:*-Find, by trial, two numbers, as near the true number as possible, and operate with them as in the question; marking the errors which arise from each of them.

Multiply the difference of the two nun:bers, found by trial, by the least error, and divide the product by the difference of the errors, when they are alike, but by their sum when they are unlike.

Add the quotient, last found, to the number belonging to the least error, when that number is too little, but subtract it when too great, and the result will give the true quantity sought.

EXAMPLES.

1. A son asking his father how old he was, received this answer: Your age is now one fourth of mine; but 5 years ago, your age was only one fifth of mine. What then are their two ages? Ans. 20 and 80.

2. A workman was hired for 30 days, at 2s. 6d. per day, for every day he worked; but with this condition, that for every day he played, he should forfeit 1s. Now it so happened, that upon the whole he had 24. 14s, to receive. How Ans. 24. many of the days did he work?

3. A and B began to play together with equal sums of money: A first won 20 guineas, but afterwards lost back of what he then had; after which, B had 4 times as much as A. What sum did each begin with? Ans. 100 guineas. 4. Two persons, A and B, have both the same income. A saves of his; but B, by spending 50% per annum more than A, at the end of 4 years finds himselt 1007 in debt. What does each receive and spend per annum?

Ans. They receive 1257. per annum; also A spènds 1007., and B spends 1504, per annum.

For since, by the supposition, r: s:: x-a: xb, therefore by division, r — s ; s :: b— @ -6, which is the 2d rule.

PRACTICAL QUESTIONS IN ARITHMETIC.

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1. THE swiftest velocity of a cannon-ball is about 2000 feet in a second of time. Then in what time, at that rate, would such a ball move from the earth to the sun, admitting the distance to be 100 millions of miles, and the year to contain 365 days 6 hours? Ans. 8 years. 2. What is the ratio of the velocity of light to that of a cannon-ball, which issues from the gun with a velocity of 1500 feet per second; light passing from the sun to the earth in 84 minutes ? Ans. the ratio of 704000 to 1.

3119

3. The slow or parade-step being 70 paces per minute at 28 inches each pace, it is required to determine at what rate per hour that movement is?

Ans. 1 miles.

4. The quick-time or step in marching, being 2 paces per second, or 120 per minute, at 28 inches each, at what rate per hour does a troop march on a route, and how long will they be in arriving at a garrison 20 miles distant, allowing a halt of one hour by the way to refresh?

Ans. The rate is 3 miles an hour, and the time 74 hours, or 7 hours 174 min.

5. A wall was to be built 700 yards long in 29 days. Now, after 12 men had been employed on it for 11 days, it was found that they had completed only 220 yards of the wall. It is required to determine how many men must be added to the former, that the whole number of them may just finish the wall in the time proposed, at the same rate of working?

Ans. 4 men to be added. 6. Determine how far 500 millions of guineas will reach, when laid down in a straight line touching one another; supposing each guinea to be an inch in diameter, as it is very nearly? Ans. 7891 miles, 728 yds., 2 ft. 8 in.

7. Two persons, A and B, being on opposite sides of a wood, which is 536 yards about, begin to go round it, both the same way, at the same instant of time; A goes at the rate of 11 yards per minute, and B 34 yards in 3 minutes; the question is, how many times will the wood be gone round before the quicker overtake the slower? Ans. 17 times.

8. A can do a piece of work alone in 12 days, and B alone in 14; in what time will they both together perform a like quantity of work?

Ans. 6 days. . 9. A person who is possessed of a share of a copper-mine, sold of his interest in it for 18007.; what was the reputed value of the whole at the same rate? Ans. 40007.

10. A person, after spending 207. more than of his yearly income, had then remaining 301. more than the half of it; what was his income? Ans. 2001. 11. The hour and minute-hands of a clock are exactly together at 12 o'clock; when are they next together? Ans. 1 hr., or 1 hr. 5 min. 12. If a gentleman, whose annual income is 1500l., spend 20 guineas a-week; whether will he save or run in debt, and how much in the year ?

Ans. Save 4081.

13. A person bought 180 oranges at 2 a penny, and 180 more at 3 a penny; after which he sold them out again at 5 for 2 pence; did he gain or lose by the bargain? Ans. He lost 6 pence.

14. If a quantity of provisions serves 1500 men 12 weeks, at the rate of 20 ounces a-day for each man; how many men will the same provisions maintaiu for 20 weeks, at the rate of 8 ounces a-day for each man? Ans. 2250 meu.

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