number, viz. 8, and also by one fourth, viz. 6. Whole amount, 74.

Now it is evident that we have not supposed the right number, otherwise the amount would have been 185, as given in the sum. We have, however, increased the number we supposed, viz. 24, by the same or similar additions, as the teacher did the true number of his scholars; consequently, 74, the number we obtained, must have the same ratio to 24, the number assumed, as 185 has to the real number of scholars in the school. Therefore, 74: 24: 185: the number required; viz. 60. Proof, 60+60+30+20+15=185.

We have then the following rule:

RULE.-Take any convenient number and proceed with it according to the conditions of the question, and observe the result; then say, as the number thus obtained is to the given number, so is the assumed number to the true one. Or the numbers may be canceled by arranging the terms as directed in Simple Proportion.

1

Ex. 2. A man being asked how much money he had, replied, that §,,, of his money added, made $57. How much had he? Ans. $60.

money

3. What number is that, which, being multiplied by 9 and divided by 4, the quotient will be 27? Ans. 12.

4. A man borrowed a sum of money on interest, which in 10 years amounted to $1800 at 6 per cent.; what was the sum? Ans. $1125.

5. Two boys were playing at marbles. Says one to the other, 1, 4, and of my marbles added together make 45, and if you can now tell how many I have, you may have them. How many had he?

Ans. 60.

6. A boy wishing to try the skill of his companions in figures, said he had a pile of apples, of which, if he gave to A., to B., and to C., there would remain 28 for D.; and requested them to tell him how many there were in all. was the number? Ans. 112.

What

years

7. A person being asked his age, said that if of the he had lived were multiplied by 7, and of them added to the product, the sum would be 292. How old was he? Ans. 60 years.

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 ?

DOUBLE POSITION.

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 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?

Suppose, 1st, that C. had $15

then B. had by the conditions, 25 and A.,

30

$70, a sum of money less than

that found; therefore, $80-$70-$10, 1st error.

$85, a sum of money greater

than that found; therefore, $85-$80-$5, 2d error.

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 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 erret.

Here both errors are too small, therefore, 6×20=120; and 8×30=240; then, 240-120-120, the difference of the products; and 30-2010, the difference of errors. Therefore, 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.

A.

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

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.

and C. $110.

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; 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 How many were there of each kind? Ans. 5 oxen,

COW.

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?

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. 30. 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.