2. Then say, as the difference of that simple, whose quantity is given, is to the known quantity, :: so is any other difference, to the quantity of its oppo

site name..

EXAMPLES.

1. How much sugar, at 9, 10, 11, and 12 cents per lb. must be mixed with 50lb., at 20 cents per lb. to be worth 12 cents per lb.?

Proof of the above as follows, viz.

9=4213lb. at 9cts.

467x4+50=237.5)2968.75(12.5 Proof.

2375

5937

4750

11875

11875

2. How much malaga, at $1.124 per gallon, sherry, at 871, white wine, at 621, must be mixed with 30 gallons of canary, at 75 cents per gallon, so that the mixture may stand in 682 cents per gallon?

Ans. 30 gal. at 1124, 30 at 872, and 330, at

62 cents per gallon.

3. How much alloy, and how much gold, of 21, and 23 carats fine, must be put to 30 ounces, of 20 carats fine, to bring it to 18 carats fine?

ALLIGATION TOTAL.*

Ans.

Alligation total is a method, when the particular rates of all the ingredients proposed to be mixed, the sum of all their quantities, with the mean rate of that sum being given, to find the particular quantities of the mixture.

RULE.

Set down all the particular rates, with the mean rate, as before; find the differences, add all the differences into a sum; then say, as the sum of all the differences is to the sum of all the quantities given, so is each particular difference, to its particular quantity required..

EXAMPLES.

1. A grocer has three kinds of rum; one at 50, one at 60, and one at 70 cents per gallon, and has a mind to put up a cask of 30 gallons of the three mixed, so

In

*Alligation Total, so called, on account of so much being given in the problems belonging to this case. fact, the total, or whole matters are given, to find the particular quantities of the mixture, or of which it is composed.

150.

that it may be worth just 564 cents per gallon; how much of each sort must he take?

133+32=173) 171+6+61: 30 :: 174: 173

561 60

61

30:30 :: 64: 64

30:30 :: 61: 64

Stated quantity, 30)16.875(56.25cts.stated price. 2. A druggist has simples, at 18, 15, 12, 9, 8, and 5 cents per lb.; how much of each sort must he take, to make up 3186. worth 11cts, per lb.

3. Hiero, king of Syracuse, gave orders for a crown to be made, entirely of pure gold; but suspecting the workmen had debased it, by mixing with it silver or copper, he recommended the discovery of the fraud to the famous Archimedes, and desired to know the exact quantity of alloy in the crown.

Archimedes, in order to detect the imposition, procured two other masses, the one of pure gold, and the other of silver or copper, and each of the same weight with the former; and by putting each separately into a vessel full of water; the quantity of water expelled by them, determined their specific bulks. From which, and their given weights, it is easier to determine the quantities of gold and alloy in the crown by this case of Alligation, than by an algebraic process.

Suppose the weight of each mass to have been 5lb., the weight of the water expelled by the alloy, 23 ounces, by the gold, 13 ounces, and by the crown, 16 ounces; that is, that their specific bulks were as

23, 13, and 16; then what were the quantities of gold and alloy in the crown.

16

And the sum of these is 7+3: 10, which should have been but 5.

13,7 of gold, ) 23/3 of alloy. Wherefore by the rule,

57: 3116. of gold.} Ans.

10: 5 :: 23: 14lb. of

4. A cask of 58 gallons is filled with liquor of 7, 8, and 10 cents the gallon, and then it stands in 94cts. the gallon; I would know how many gallons of each sort was taken. Ans. 40 gal. at 10cts. 8 at 8cts. and 8

gal.

at 7cts.

SINGLE POSITION,

OR THE RULE OF FALSE,

Is so called, because by supposed numbers, taken at adventure, and worked with according to the nature of the question, the true number sought is discovered.

Take any number, and perform the same operations with it, as are described to be performed in the question, by the following

RULE.*

As the sum total of the errors,

Is to the given sum,

So is the supposed number,

To the true one required.

* If we put s=sum total of errors, S=given sum, n= supposed number, and N=true number; then by the na

ture of the rule, s: S::n:

Sn

=N Ans.

S

EXAMPLES,

of his

1. A teacher said that the 4, 3, 4, 1, and scholars amounted to 87; how many students had he? Suppose he had 1 scholar, then, per question, ++++ of 1-2, therefore,

29

=

29

20: 1 :: 87 : 60 Ans.

From this operation it appears, that the result of the position, bears the same proportion to the position, that the given number does to the number required, which is a principle to be observed throughout the rest of this case.

2. Four merchants, A, B, C, and D, have gained $1000, which they divide in manner following, viz. That half the share of A, is severally equal to the share of B, the share of C, wanting $11, and the share of D, plus $11; required the share of each.

5

$1000 proof.

1

3. A company of students drank at a reckoning a certain sum, and said that, 4, 4, and of the money made exactly $178; required, their reckoning. Ans. $20.

4. A man dying, left $10000 to be divided amongst his three sons, (whose ages were 19, 16, and 10 years respectively) in such a manner that their several portions, when they became of age, might be equal, reckoning simple interest at 6 per cent, during their minorities; required, the share of each.