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 each person's age?

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; the third ; and the fourth the remainder, which is 281.: what is the sum? Ans. 1121.

3. A person, after spending and of his money, had 601. left: what had he at first?

Ans. 1441.

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

5. A person bought a chaise, horse, and harness for 601.; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness: what did he give for each?

Ans. 11. 6s. 8d. for the horse, 61. 13s. 4d. for the harness, and 401. for the chaise.

6. A vessel has 3 cocks, A, B, and C; A can fill it in 1 hour, B in 2, and C in 3: in what time will they all fill it together? Ans. hour.

DOUBLE POSITION.

Double Position teaches to resolve questions by making two suppositions of false numbers.

RULE.*

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

* The rule is founded on this supposition, that the first error is to the second, as the difference between the true and first sup¬ posed 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 the supposition, may be thus demonstrated.

Let A and B be any two numbers, produced from a and b by similar operations: it is required to find the number, from which is produced by a like operation.

Put x = number required, and let No1=r, and N—B=s. Then according to the supposition, on which the rule is founded, r: s :: x—a : x-b, whence, by multiplying means and extremes, rx-rb=8x-sa; and, by transposition, rx—sx=rb -8a; and, by division, ♬ =

rb-sa

number sought.

Again, if r and s be both negative, we shall have —r : ::: x—ɑ : x—b, and therefore -rx+rb=—sx+sa; and rx—

rb-sa

sx=rb-sa; whence x:

as before.

r

rb+sa, r+s

In like manner, ifr or s be negative, we shall have x=

by working as before, which is the rule.

NOTE. It will be often advantageous to make 1 and 0 the suppositions.

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

in the question.

3. Multiply each of the errors by the contrary supposition, and find the sum or difference of the products.

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

5. If the errors be unlike, divide the sum of the products by the sum of the errors, and the quotient will be the an

swer.

NOTE. The errors 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 tabby at 4s. a yard, and Persian at 2s. a yard; the whole number of yards she bought was 8, and the whole price 20s.: how many yards had she of each sort?

Suppose 4 yards of tabby, value Then she must have 4 yards of Persian, value

16s.

8

Sum of their values 24

So that the first error is + 4

Again, suppose she had 3 yards of tabby at 12s.
Then she must have 5 yards of Persian at 10

Sum of their values 22

So that the second error is + 2

Then 4-2-2= difference of the errors.

Also 4x2=8= product of the first supposition and sec

ond error.

And 3×4=12= product of the second supposition by the

first error.

And 12-8=4= their difference.

Whence 4-2-2- yards of tabby, } the answer.

And 8-2-6 yards of Persian,

2. Two persons, A and B, have both the same income; A saves of his yearly; but B, by spending 50l. per annum more than A, at the end of 4 years finds himself 1001. in debt: what is their income, and what do they spend per annum?

Ans. Their income is 1251. per annum; A spends 1001. and B 1501. per annum.

3. Two persons, A and B, lay out equal sums of money in trade; A gains 1261. and B loses 871. and A's money is now double that of B: what did each lay out? Ans. 3001.

4. A laborer was hired for 40 days, on this condition, that he should receive 20d. for every day he wrought, and forfeit 10d. for every day he was idle; now he received at last 21. 1s. 8d.: how many days did he work, and how many was he idle?

Ans. He wrought 30 days, and was idle 10.

5. A gentleman has two horses of considerable value, and a saddle worth 501.; now, if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the first: what is the value of each horse? Ans. One 301. and the other 401.

6. There is a fish, whose head is 9 inches long, and his tail is as long as his head and half as long as his body, and his body is as long as his tail and his head: what is the whole length of the fish? Ans. 6 feet.

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.

This is also called Variation, Alternation, or Changes; and the only thing to be regarded here is the order they stand in; for no two parcels are to have all their quantities placed in the same situation.

The Combination of Quantities is the showing how often a less number of things can be taken out of a greater, and combined together, without considering their places, or the order they stand in.

This is sometimes called Election, or Choice; and here every parcel must be different from all the rest, and no two are to have precisely the same quantities, or things.

The Composition of Quantities is the taking a given number of quantities out of as many equal rows of different quantities, one out of each row, and combining them together.

Here no regard is had to their places; and it differs from combination only, as that admits of but one row, or set of things.

Combination of the same form are those, in which there is the same number of quantities, and the same repetitions : thus, abcc, bbad, deef, &c. are of the same form; bụt abbc, abbb, aacc, &c. are of different forms.