8. A merchant has 6lbs. of tea at 9s. a pound, and 9lbs. at 14s., which he would mix with some at 8s., and some at 13s.; how much of these two last must he take that the mixture may be worth 11s. a pound?

It is obvious, that we first find the mean rate of the simples whose quantities are given, and then proceed as before. Ans. 15lbs, of each. 9. How much gold of 15 and 17 carats fine, must be mixed with 5oz. of 18, and 13oz. of 22 carats fine, that the compound may be 10 carats fine? Ans. 2oz. of each. 10. How much wine at 80cts. and at 87 cts. per gallon, must be mixed with 8 gallons at 75cts, and 12 gallons at 90cts. a gallon, that the mixture may be worth 82 cts. a gallon?

Ans. 52 gallons at 80cts., and 20 gallons at 871

221. When the quantity of the compound is given, RULE. As the sum of the RELATIVE QUANTITIES, found by the FOREGOING RULE is to the quantity REQUIRED: so is each QUANTITY SO FOUND: to the REQUIRED quantities of EACH.

1. A merchant has tobacco at 10cts., at 12cts., and at 14cts. a pound, of which he would sell A 100lbs. at 13cts. a pound; how much of each kind must he take?

We find the relative quantities to be 3lbs. at 10cts., 3lbs. at 12cts. and 4lbs. at 16cts.; the sum of which is 3+3+4=10lbs.

2. A grocer has currants at 8d., 12d., 18d. and 22d. a pound; the poorest will not sell, and the best are too dear; he therefore concludes to make a mixture of 120lbs., and so much of each sort as to sell the mixture at 16d. a pound; how much of each must he take?

Ans. 36lbs. at 8d., 12lbs. at 12d., 24lbs. at 18d., and 48lbs. at 22d. 3. A retailer has by him four sorts of tea, viz., at 5s., 6s., 8s. and 9s. a pound; out of these he is ordered to make up a chest, containing 168lbs., so as to be worth 7s. a pound; what quantity of each must he take?

4. A merchant would form a compound of 72 gallons of wine, of some at $.75, $1.25, $1.50, and $1.62 a gallon, so as to sell it at $1.374; what quantities of each must he take?

Ans. 8 gallons at $ .75, 16 at $1.25, 40 at $1.50, and 8 at $1.62§. 5. A silversmith has silver of 11 oz. fine, and of 7oz. fine, and has orders to make up a piece of work, requiring 35lbs. of 94oz. fine; how much of each must he take. Ans. 21lbs. 4oz. 13pwt. 8grs. of 114oz. fine; and 13lbs. 7oz. 6pwt. 16grs. of Toz. fine.

6. Suppose 9lbs. of pure gold immersed in a vessel full of water to expel 3lbs. of water, 9lbs. of pure silver to expel 6lbs, and 9lbs. of a mass made up of gold and silver, to expel 4lbs.; required the quantities of gold and silver in said mass? Ans. 6lbs. of pure gold, and 3lbs. of silver.

7. Hence is solved the curious question concerning King Hiero's crown.-Hiero, king of Sicily, ordered a crown to be made containing 63oz. of pure gold; but suspecting the workmen had debased it by substituting part silver therein, he recommended the detection of the fraud to the famous Archimedes, who putting it into a vessel of water found it raised the fluid, or that itself contained 8.2245 cubic inches of metal, and having discovered that the cubic inch of gold more critically weighed 10.36 ounces, and that of silver but 5.85 ounces, he found by calculation what part of his Majesty's gold had been changed; and you are desired to repeat the process. Ans. 34.1964oz. of pure gold, and 28.3036oz. of silver.

THEORETICAL QUESTIONS.

What is alligation? alligation medial? the rule? What is alliga tion alternate? When the price of the mixture, and the price of the several simples, are given, how do you find the relative quantities of EACH? When the quantity of one simple is given, how do you find the others? When the quantity of the whole compound is given, how do you find the quantity of EACH SIMPLE ?

POSITION.

222. Position is a rule, that by FALSE or SUPPOSED num bers, taken at pleasure, discovers the TRUE one required. It is of two kinds, SINGLE and DOUBLE.

SINGLE POSITION.

223. SINGLE POSITION is that which makes use of but ONE supposed number.

* *

* * *

224. RULE.* Take ANY convenient number, and proceed with it as though it were the true one; then say, as the RESULT is to the GIVEN SUM:: so is the supposed number: to the number required.

1. A person being asked how old he was, answered, "if to my age you add of it, and of it, the sum will be 44;" required his age? Suppose his age was 18

years; then

of it, would be

6

4 of it,

9

Then, as the RESULT 33: 44 ::

Ans.

Given Sum. Sup'd. Number. Number Req'd.
18
: 24,
of his age; this added

BETTER ANALYTICALLY. of his age and of his age = to his age by the question is equal to 44; that is, +(=)=44. Hence 4424 years, Ans.

2. A school-master, being asked how many scholars he had, replied, "if of the number I have be multiplied by 7, and of this product be divided by 3, the quotient will be 20;" what number of scholars had he? Ans. 30.

The same analytically. If the quotient by dividing by 3 is 20, the dividend is 3 x 20 = 60; this by the question, is equal to 5 of the product of of the numbers multiplied by 7; consequently, 60 ÷ 5 = 84 = the product of of the number by 7; and 84 ÷ 7 = 12 of the number. Hence 12÷ 2 =30, Ans.

* Questions that belong to this rule are such, that the number sought is multiplied or divided by some known number; or is increased or diminished by itself a certain number of times. Consequently the results will be proportional to the suppositions; and hence the reason of the rule is obvious:

Result.

Result.

Nos. from which each result is obtained.

3. A vessel has 3 pipes; the first can fill it in an hour, the second in of an hour, and the third in of an hour; in what time will all three fill it together?

Ans. 1 hour. By analysis. It is obvious, that the first will fill 2 in 1 hour, that the second will fill 3 in one hour, and that the third will fill 4 in one hour; Consequently all three will fill 2+3+4= 9 in 1 hour; hence they will all fill 1 in 1 of an hour, Ans. 4. A gentleman distributed 78 cents among a number of poor people, consisting of men, women, and children; to each man he gave 6 cents, to each woman 4, and to each child 2; now there being twice as many women as men, and thrice as many children as women, it is required to find the number of each? Ans. 3 men, 6 women, and 18 children.

5. A, B & C, talking of their ages, B said his age was once and a half the age of A, B said his was twice, and one tenth the age of both, and that the sum of their ages was 98; what was the age of each?

Ans. A's 12, B's 18, and C's 63.

6. There is a cistern with 3 unequal pipes; if the largest pipe be opened it will be empty in 1 hour; if the second be opened it will be empty in 2 hours; if the third be opened it will be empty in 3 hours: required the time it will take to empty if all run together? Ans. 328 minutes.

7. A, B, C and D spent 358. at a reckoning, and, being a little dipped, they agreed that A should pay 3, B, C, and D 4: what must each pay in such proportion? Ans. A 13s. 4d., B 16s., C 6s. 8., and D 5s. 8. Peter drinks a barrel of beer in 24 days, Charles when he goes about it, does it in 16 days; now if they should both drink together, in what time would they finish it?

Ans. 9d. 14h, 24m. 9. A gentleman bought a chaise, horse and harness for $500; the horse cost more than the harness, and the chaise more that the horse; what was the price of each?

Ans. chaise $212.76545, horse $159.57422, harness $127.659.

10. A wolf says he can eat a sheep in 60 minutes, a bear says he can eat it in half the time, a lion says he can eat it as quick as both; in what time can all three eat it together?. Ans. 10 minutes.

11. A gentleman divided his fortune among his sons, to A he gave $9 as often as to B $5, and to C but $3 as often as to B $7, yet C's portion came to $1059; what was the whole estate? Ans. $7979.80.

DOUBLE POSITION

225. Is that which makes use of two supposed numbers.

** * * * * * * * * * * * * 226. RULE* I. Take ANY Two convenient numbers, and proceed with EACH as though it was the TRUE ONE required.

II. Place the RESULTS OF ERROURS against their positions or supposed numbers; and if any errour be TOO GREAT, mark it with; but if TOO SMALL with

III. Multiply them crosswise; that is, the first position by the last errour, and the last position by the first errour.

IV. Then if the errours are ALIKE, divide the DIFFERENCE of

The rule is founded on this supposition, that the first errour is to the second as the difference between the true and first supposed number, is to the difference be tween the true and second supposed number. Where this is not the case, therefore. exact answer to the question cannot be found by this rule.

the products by the DIFFERENCE of the errours; but if the errours are UNLIKE, divide the sum of the products by the sum of the errours.

1. B asked C how much his horse cost; C replied, that if he cost him 3 times as much as he did and 15 dollars more, he would stand him in $300; what was the price of the horse?

Better analytically. It is obvious, that, taking the question backwards, 300-153 will give the cost of the horse. 300-15 = 285, and 285395, Answer, as before.

2. A and B have the same income; A saves of his, but B, by spending $75 per annum more than A, at the end of 8 years finds himself $100 in debt; what is their income, and what does each spend yearly?

Ans. their income is $500; A spends $437, and B $5124. The same by analysis. On an average, it is evident, that B falls in debt 12 dollars every year; by the question, therefore, it is plain that of their income +$75 is equal to of it +$124; consequently, of their income is equal to $75-121 ($621.) Hence their income is 8 times $621 $500, Answer, as before.

3. A man was hired for 50 days on these conditions, that for every day he worked, he should receive $.75, and, for every day he was idle, he should forfeit $ .25; at the expiration of the time, he received $27.50; how many days did he work, and how many was he idle?

Ans. he worked 40 days, and was idle 10. By analysis. Had he worked the whole time, his wages would evidently have been $ .75 × 50 = $37.50, which is $10 more than he did receive; but every day he was idle lessened his wages $75+ $25 = $1; consequently he was idle 10 days, Answer, as before.

4. A man had two silver cups of unequal weight, having one cover to both, which weighs 5oz.; now if the cover is put on the less cup it will be double the weight of the greater, and if put on the greater, it will be three times as heavy as the less; required the weight of each? Ans. 3oz. less-and 4oz. greater. 5. There is a fish whose head is 9 inches long, his tail as long as his head and half his body, and his body as long as his head and tall both; what is its whole length? Ans. 6 feet. 6. A person being asked, in the afternoon, what o'clock it was, answered, that the time past from noon was equal to 2 of the time to midnight; required the time? Ans 36m. past one.

7. A & B, settling accounts found, that if £6 were added to bill, and the same sum taken from of B's, the sum would be

of A's

of the

remainder; and that the sum and remainder added together made £72 pounds; what was each man's bill?

227. PERMUTATION is finding how many different ways the order of things may be varied or changed.

1. Four persons agreed to board together so long as they could sit in different position at the table at dinner; pray, how long did they tarry?

If there had been Two persons, a & b, they could sit in (1 X 2 =) 2 positions; a b and ba; had there been THREE, they could sit in (1 x 2 x 3 =) 6 positions; for beginning with each one, we shall have Two positions EACH time; thus, a b c, acb; bac, bca; and c a b, c ba, In like manner, if there be FOUR, the different positions will be 1 × 2 × 3 × 4 = 24. Ans. 24 days.

228. Hence, to find the number of different changes or permutations of which any number of different things are capable,

RULE. Multiply together all the terms of the natural series of numbers, from 1 up to the given number, and the last product will be the answer.

2. Christ Church in Boston has 8 bells; how many changes may be rung on them? Ans. 40320.

3. How many variations will the nine digits admit of? Ans. 362880. 4. How many changes or variations may be made of the 26 letters of the alphabet ? Ans. 620448401733239439360000. 229. COMBINATION is finding how often a less number of things can be chosen from a greater.

We have seen; that any 3 things, a, b, c, all different from each other, admit of 6 variations; but if the things are all alike, as a, a, a, then the 6 variations will be 1X 2X3 reduced to 1, which may be expressed thus, = 1. Again, if but two 1X2 X3 things out of three are alike, as a, a, c; then the 6 variations will be reduced to 2e, ca a, ac a, which may be expressed thus,

2 x 3

X

2

3. And so on.

e, To find the COMBINATIONS of any given number of Ferent from each other, taking any given number at a time,