8. A. saves of his income, but B. who has the same income, spends twice as fast as A., and thereby contracts a debt of $120 annually. What is their income? Ans. $360. 9. The sum of A., B., and C's ages, is 132 years. B's age is 1 the age of A; and C's age is twice as great as B's. What are their respective ages? Ans. A's age is 24; B's, 36; and C's, 72 years.

QUESTIONS.-What is Position? What is Single Position? What is the rule? How may the operations be canceled ?

By Double Position, we solve such sums as require two suppositions.

In this rule, the numbers supposed to be the true ones bear no certain or definite proportion to the required answers.

RULE.-Assume any two convenient numbers and proceed with each according to the conditions of the question, and compare the result of each with the sum or result given in the question, and find their differences. Call each difference an error.

Multiply the first assumed number by the last error; and the last assumed number by the first error.

If both errors are too great or too small, divide the difference of these products by the difference of the errors, and the quotient will be the number sought. But if But if one of the errors be too large, and the other too small, divide the sum of the products by the sum of the errors.

Note. The errors are said to be too large or too small, when by operating on each supposed number according to the nature of the question, the number obtained is greater or less than the corresponding number in the sum.

Ex. 1. Three men found a purse of money containing $80,

which they agree to divide in such a manner, that A. shall have $5 more than B, and that B. should have $10 more than C. What was each man's share of the money?

If now the above operations be compared with the rule and the note following, it will be seen that the first error is too small, and the last one too large; therefore, 15, number first supposed × 5, the last error=75; and 20, the number last supposed x 10, the first error=200; and 200+75=275, the sum of the products; and 10+5=15, the sum of errors. Therefore, 275÷15=$18.333+, C's share; and $18.333+$10= $28.333+, B's share; and $28.333+$5-$33.333, A's share.

2. Four individuals having $100 to divide among themselves, agree that B. should have $4 more than A.; C., $8 more than B.; and D. twice as much as C. What was each man's share?

and 100-70=30, 1st error. Hence, 100-80=20, 2d error. Here both errors are too small, therefore, 6×20=120; and 8×30-240; then, 240 120 120, the difference of the products; and 30 There-20=10, the difference of errors.

fore, 120-10=12, A's share; 12+4=16, B's share; and 16 +8=24, C's share; and 24+24=48, D's share. +16+24+48=100.

Proof, 12

3. Three men hired a piece of wall built, for which they paid $500. Of this, A. paid a certain part; B. paid $10 more than A., and C. paid as much as A. and B. both. What did each man pay ? Ans. A. paid $120; B. $130; and C. $250.

Sums like the preceding are solved with ease by analysis. Since we have the sum they all paid, we know that C. paid $250, because he has paid as much as the other two, that is, one half of the whole. Therefore, A. and B, together paid $250. But B. paid $10 more than A, hence, 250-10-240, twice the number of dollars A. paid, and 240÷2=120, A's share; then, 120+10=130, B's share; and 120+130=250, C's share.

4. Two persons lay out equal sums of money in trade. A. gains 120 £., and B. loses 80 £. A's money was then treble With what sum did they commence?

B's.

Ans. 180 £.

5. A farmer hired a laborer 40 days, on condition that he should receive 20 cents for every day he wrought, and forfeit 10 cents every day he was idle. At the expiration of the 40 days he received $5. How many days did he work, and how many was he idle? Ans. He wrought 30 days, and was idle 10 days.

6. What is the length of a fish whose head is 10 inches long, his tail as long as his head and half the length of his body, and his body as long as his head and tail both? Ans. 80 inches.

7. Two persons, A. and B., have the same income. A saves of his, but B., by spending $150 per annum more than A., at the end of 8 years finds himself $400 in debt. What was their income, and how much did each spend annually? Ans. Income, $400. A spends $300, and B. $450.

8. A man bequeathed his property to his three sons, on the following conditions; viz. to A., one half, wanting $50; to B., one third; and to C., the remainder, which was $10 less than B's share. How much did each son receive, and what was the whole estate? Ans. A. received $130; B. $120; and C. $110. The whole estate was $360.

9. A farmer bought a certain number of oxen, cows, and calves; for which he paid 130 £. For every ox he paid 7£.; for every cow, 5 £.; and for every calf, 1£. 10s. There were two cows for every ox, and three calves for every CÓW. How many were there of each kind? Ans. 5 oxen, 10 cows, and 30 calves.

10. A person after spending $10 more than

of his annual

income, had $35 more than of it remaining. What was his income? Ans. $150.

11. A person has two horses; he also has a saddle worth 10 £. If the saddle be placed on the first horse, the horse and saddle are worth twice as much as the second horse; but the value of the second horse with the saddle is 13 £. less than the value of the first horse. How much is each horse worth? Ans. The first is worth 56 £., and the second, 33 £.

QUESTIONS.-What is Double Position?

What relation do the

supposed numbers bear to the true ones? What is the rule? When are the errors said to be too large or too small?

JOUS

PROMISCUOUS EXAMPLES.

Ex. 1. If 460 be multiplied by 36, and the product divided by 9, what will the quotient be? Ans. 1840.

2. What number is that which, when increased by of itself, will be 126? Ans. 72.

3. What number multiplied by will produce 16? Ans. 211.

91.

4. What fraction multiplied by 15 will produce? Ans. 3. 5. What number multiplied by 32 will produce 2912? Ans.

6. What number divided by 21 will give 65 as a quotient? Ans. 1365.

7. How many nails are required to shoe 27 horses, each shoe requiring 8 nails? Ans. 864.

8. In the counter of a merchant there are four drawers, in each drawer, 4 divisions, and in each division, $23.75. How many dollars do the four drawers contain? Ans: $380.00.

9. Two men depart from the same place and travel the same way; one travels 36 miles per day, and the other 42. What will be the distance between them at the end of the 8th day, and how far will each have traveled? Ans. 48 miles apart, the one having traveled 288, and the other 336 miles.

10. A person owning of a ship, sold of his share for $474. What was the value of the whole ship, at the same rate? Ans. $1264.

11. How many men must be employed to finish a piece of work in 15 days, which would require 5 men 24 days? Ans. 8 men.

12. A person being asked the time of day, answered, “The time past noon is equal to the time till midnight." What was the time? Ans. 36 minutes past 5.

13. In a certain school, the scholars learn to read and write; learn geography; learn grammar; and 16 study astronomy. What was the number in the school? Ans. 128.

14. What is the whole length of a pole, of which stands in the ground, 16 feet in the water, and in the air? Ans. 213 feet, 4 inches.

15. There is a room 12 feet long, 8 feet wide, and 7 feet high. How much paper, 2 feet wide, will be required to paper the same? Ans. 46 yards, 2 feet.

16. My horse and saddle are both worth 36 £. 12 s., and my horse is worth 7 times as much as my saddle. What is the value of each? Ans. My horse is worth 32 £. 0 s. 6 d., and my saddle, 4 £. 11 s. 6 d.

17. There is a cistern having 3 faucets, the largest of which will empty it in one hour, the second in 2 hours, and the third in 3 hours. In what time will they all empty it, if opened at the same time? Ans. 32 minutes.

18. Divide 1500 acres of land between A., B., and C., so that A. shall have 150 acres more than B., and B. 100 acres more than C. Ans. A. has 633, and B. 483, and C. 383.

19. A certain pasture will feed 324 sheep 7 weeks. How many must be turned away, in order that it may be sufficient for the remainder 9 weeks? Ans. 72.

20. A merchant bought 120 gallons of melasses for $45. How must he sell the same per gallon, to gain 15 per cent.? Ans. 0.43.+

21. If a family of 8 persons consume $200 worth of provision in 9 months, how much will 18 persons consume in a year? Ans. $600.

22. A man left his son a fortune, of which he spent in 3 months; and in 6 months more he spent of the remainder, when he had only $1500 remaining. What was his fortune? Ans. $13500

23. A young man received 350 £. as his share of his fa