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Case 3.- When one of the ingredients is limited to a cer

tain quantity. RULE.—Take the difference between each price and the mean rate as before; then, as the difference of that simple whose quantity is given, is to the rest of the differences severally, so is the quantity given to the several quantities required.

EXAMPLES. 1. A grocer would mix teas at 1 dollar 20cts. 66cts. and 1 dollar per pound, with 20 pounds at 40 cents per pound; how much of each sort must he take to make the composition worth 80 cents per pound:

16 t t to s 407 40

66 | 20 40:20::20:10 at 66cts. ) ou 100 14 40:14::20: 7 at $1 Ans.

| 120 j. 40 40:40::20:20 at 1,20 ) 2. How much wine at 80cts. at 88 and 92 per gallon, must be mixed with 4 gallons at 75cts. per gallon, so that the mixture may be worth 86 cents per gallon ?

Ans. 4gals. at 8Octs. 81 at 88 and 8 at 92. 3. With 95 gallons of rum at 8s. per gallon, I 'mixed other rum at 6s. 8d. per gallon, and some water; then I found it stood me in 6s. 4d. per gallon ;-I demand how much rum at Os. 8d. I took, and how much water.

Ans. 95 gallons rum at (s. 8d. and 30 gallons water.

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POSITION.

Position is a method of performing such questions as cannot be resolved by the common direct rules, and is of two kinds, Single and Double.

SINGLE POSITION. Single Position teaches to resolve those questions whose results are proportioned to their suppositions.

RULE.-1. Take any number and perform the same operations with it, as are described to be performed in the question,

2. Then say, as the result of the operation is to the position, so is the result in the question to the number required.

EXAMPLES. 1. A's age is double that of B, and B's is triple that of C, and the sum of all their ages is 140; what is the age of each ? Suppose A's age to be 48

Then will B's=48=24

And C's=~= 8

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140 Proof. 2. A certain sum of money is to be divided between 4 persons in such a manner that the first shall have { of it, the second 4, the third ļ, and the fourth the remainder, which is 28 dollars; what is the sum ? Ans. $112.

3. A person, after spending { and I of his inoney, had 60 dollars left; what had he at first? Ans. $144.

4. What number is that which being increased by 4, and 4 of itself, the sum will be 125 ? Ans. 60.

5. A person lent his friend a sum of money, to receive interest for the same at 6 per cent. per annum, simple interest; at the end of three years he received for principal and interest 383 dollars 50 cents ; what was the sum lent?

Ans. 325 dollars. 6. A cistern is supplied with three cocks, A, B, and C: A can fill it in 1 hour, B in 2, and C in 3; in what time will it be filled by all of them together? Ans. If hour.

DOUBLE POSITION. DOUBLE POSITION teaches to resolve questions by making two suppositions of false numbers.*

* Questions in which the rusults are not proportional to their prisitions, belong to this rule; such are those, in which the number sought is increased or diminished by some given number, which is no known part of the number required.

RULE.-1. Take any two convenient numbers, and proceed with each according to the conditions of the question.

2. Find how much the results are different from the result in the question.

3. Multiply each of the errours by the contrary supposition.

4. If the errours be alike, divide the difference of the products by the difference of the errours, and the quotient will be the answer.

5. If the errours be unlike, divide the sum of the products by the sum of the errours, and the quotient will be the answer.

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

EXAMPLES. 1. A lady bought cambric for 40 cents a yard, and India cotton at 20 cents a yard; the whole number of yards she bought was 8, and the whole cost 2 dollars; how many yards had she of each sort ?

Suppose 4 yards of cambric, value $1,60 cts. Then she must have 4 yards of cotton, value 80

Sum of their values, 2,40

So that the first errour is +40 Again, suppose she had 3 yards of cambric, $1,20 cts. Then she must have 5 yards of India cotton 1,00

Sum of their values, 2,20

So that the second errour is + 20
Then 40—20=20=difference of the errours.
Also 4x20=80=product of the first supposition and

second errour.
And 3x40=120=product of the second supposition

and first errour.
And 120_80=40=their difference.
Whence 40+20=2 yards of cambric,

' Ans.
And 8-2=6 yards of India cotton.

2. A and B have both the same income; A saves of his yearly; but B, spending 50 dollars a year more than

A, at the end of 4 years is 100 dollars in debt; what is their income, and what do they spend per annum ?

(Their income is $125 per year. Ans. A spends $100.

(B spends $150. 3. A laborer was hired for 40 days upon these conditions, that he should receive 2 dollars for every day he wrought, and forfeit 1 dollar for every day he was idle; at the expiration of the time he was entitled to 50 dollars ; how many days did he work, and how many was he idle ?

Ans. He wrought 30 days, and was idle 10. 4. A man had 2 silver cups of unequal weight, with 1 cover for both, weight 5oz. ; now if he put the cover on the less cup, it will be double the weight of the greater ; and put on the greater cup, it will be three times the weight of the less cup; what is the weight of each cup ?

Ans. 3oz. the less, and 4oz. the greater. 5. A person being asked what o'clock it was, answered that the time past from noon was equal to of. the time to midnight; required the time.

Ans. 36 minutes past 1. 6. There is a fish whose head is ten feet long; his tail. is as long as his head and half the length of his body, and his body is as long as his head and tail; what is the · whole length of the fish ?

Ans. 80 feet. 7. A and B laid out equal sums of money in trade; A gained a sum equal to 1 of his stock, and B lost 225 dol-'. lars; then A's money was double that of B's; what did each lay out ?

Ans. $600.

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PERMUTATION AND COMBINATION.

The permutation of quantities is the showing how many different ways the order or position of any given number of things may be changed.

The combination of quantities is the showing how often a less number of things may be taken out of a greater, and combined together, without considering their pla-vy ces, or the order in which they stand.

PROBLEM 1. To find the number of permutations, or changes, that can be made of any number of things, all differing from each other.

Rule.—Multiply all the terms of the natural series of numbers, from one up to the given number, continually together, and the last product will be the answer.

EXAMPLES. 1. How many changes may be made with these three letters, A, B, C.

CHANGES. a b c 1 ) a c b 2

Por

b cald Proof.

Teens

1.

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ca b 5

. cba65 6 Answer 2. How many changes may be rung upon 6 bells ?

Ans. 720. ; 3. How many changes may be rung upon 12 bells, and · How long would they be ringing but once over, supposing 10 changes might be rung in one minute, and that the year contains 365 days, 6 hours ?

üm 479001600 changes, and 91

408, 7 years 3w. 5d. and 6 hours. 4. A young scholar coming into a town for the convenjence of a good library, demanded of the gentleman with whom he lodged, what his diet would cost for a year; he told him $150; but the scholar not being certain what time he should stay, asked him what he should give him for so long as he could place his family (consisting of 6 persons beside himself) in different positions every day at dinner ; the gentleman told him $50; to this the scholar agreed--wbat time did he stay? Ans. 5040 days.

PROBLEM II. Any number of different things being given, to find how many changes can be made out of them, by taking any given number at a time.

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