and the sum of their expenses $8+$16+$24-$48. As the ratios, in the true and supposed, are the same, it follows, that the true sum of their expenses will have the same ratio to the true expense of each individual, that the sum of their supposed expenses has to the supposed expenses of each individual.

Thus:

Suppose any number, and proceed in the operation as though it were the true; then, as the result of the operation, or sum of the errors, is to the supposed number, so is the given number to the true number required.

EXAMPLES.

2. A person, after spending and of his income, had $30 What had he at first?

left.

Suppose $60

=30 =20

60-50-10 income left:

Then 10: 60 :: 30: 180 Ans.

Or by fractions, and 3=2; then +3, the income spent, and remains=$30; then =30×6=$180, as before.

3. A certain sum of money is to be divided between 5 men, in such a manner that A. shall have, B., C., D. 26, and E. the remainder, which is $40. What is the sum?

Ans. $100.

4. A schoolmaster being asked how many scholars he had, replied, if he had as many more, and as many more, he would have 11 less than 99. How many had he? Ans. 32.

5. A man bought a horse, chaise, and harness for $216. The horse cost twice as much as the harness, and the harness one third as much as the chaise. What was the cost of the chaise ? Ans. $108.

6. What number is that whose,, 1, 1, 1, and make 127?

Ans. 90.

7. A man being asked his age, said, If you add to its double ,,, and of my age, it will be 122. What was his age?

Ans. 45.

per

8. A certain sum of money is to be divided among sons, in such a manner, that the first shall have of of; the second of 1 of 2; the third of 45; the fourth has $110. What is the sum divided ?

Ans. $240.

9. A. and B. having found a purse of money, disputed who should have it. A. said that, and of it amounted to $35, and if B. would tell him how much was in it, he should have the whole; otherwise he should have nothing. How much did the purse contain? Ans. $100.

DOUBLE POSITION.

Art. 255.-DOUBLE POSITION teaches to discover the true, by the use of two supposed numbers.

RULE.

I. Suppose two numbers, and proceed with each according to the conditions of the question, as in Single Position, noting the error. The difference between the result and the given sum is the error.

II. Multiply the first supposition by the second error, and the second supposition by the first error.

III. If the errors are alike—that is, both too great or both too small, divide the difference of the products by the difference of the

errors.

IV. If the errors are unlike—that is, one too large, and the other too small, divide the sum of the products by the sum of the errors.

OBS. This rule is founded on the supposition 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 this is not the case, the exact number cannot be obtained by this rule.

QUESTIONS.-5. What is Double Position? 6. On what supposition is this rule founded? 7. Verify the principle.

EXAMPLES.

1. A man being asked what his carriage cost, replied, If it had cost twice as much as it did, and $20 more, it would have cost $370. What was the cost of the carriage?

The foregoing question may be thus solved: Let x equal the cost of the carriage; then by the conditions of the question, 2x+20=370.

2. A., B., and C. built a house, which cost $228. B. paid

$30 more than A., and C paid as much as A. and B. What

did each pay?

A. paid $42.
B. paid $72.
C. paid $114.

3. A. and B. have the same income. A. saves of his annually, but B., by spending $120 per annum more than A., at the end of 6 years, finds himself $120 in debt. What is their income, and what does each spend annually?

Their income $400. Ans. A. spends $300, and B. spends $420.

4. A man has two silver cups of unequal weight, having one cover to both, weighing 5 oz. When the cover is put on the less cup, it weighs double the greater; when put upon the greater cup, it weighs three times the less. of each cup?

Ans.

What is the weight
The less, 3 oz.
The greater, 4 oz.

5. There is a fish whose head is 3 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 length of the fish? Ans. 24 feet.

6. A man being asked, in the afternoon, what o'clock it was, answered, that the time passed from noon was equal to of the time to midnight. Required the time.

Ans. 20 minutes past 1 o'clock. 7. A gentleman has two horses, and one carriage which is worth $100. If the first horse be harnessed into the carriage, he and the carriage together will be worth three times as much as the second horse; but if the second be harnessed into the carriage, they will be worth seven times as much as the first horse. What is the value of each horse?

Ans. $20 and $40.

8. A laborer was hired 60 days upon this condition, that for every day he wrought he should receive 3s. 4d., and for every day he was idle he should forfeit 1s. 8d. At the expiration of the time he received £3 158. How many days did he work, and how many days was he idle?

Ans.

He was employed 35 days, and was idle 25.

ALLIGATION.

Art. 260.-ALLIGATION is the method of mixing two or more simples, of different qualities, so that the composition may be of a mean, or middle quality.

When the quantities and prices of the simples are given, to find the mean price of the mixture compounded of them, the process is called

ALLIGATION MEDIAL.

Art. 261.-1. If I mix 8 lbs. of sugar, worth 10 cents a pound, with 10 lbs., worth 15 cents a pound, what is 1 lb. of the mixture worth?

Eight pounds, at 10 cents a pound, are worth 10×8=80 cents, and 10 pounds, at 15 cents, are worth 15×10=150 cents; then, 80+150-230 cents, the price of the whole mixture, and 8+10=18 pounds, the whole mixture; then $2.30 18 lbs. 123 cts., the worth of 1 pound of the mixHence the

ture.

RULE.

Multiply each quantity by its price, and divide the sum of the products by the sum of the quantities. The quotient will be the rate of the compound required.

EXAMPLES.

2. A grocer mixes sugar, 5 lbs. at 6 cts., 8 lbs. at 5 cis., and 7 lbs. at 10 cts. alb. What is 1 lb. of the mixture worth? Ans. 7 cts.

3. A farmer mixes 12 bushels of wheat at $1.75 a bushel, 8 bushels of rye at $1, and 6 bushels of corn at 80 cts. a bushel. What is a bushel of the mixture worth? Ans. $1.30.

4. A goldsmith melted together 12 lbs. of gold, 21 carats fine, 8 lbs. 20 carats fine, 9 lbs. 22 carats fine, and 7 lbs. 18 carats fine. Of what fineness is the mixture?

Ans. 20 carats fine.

5. A merchant mixed 8 gallons of wine, at 4s. 2d. per gal