2. A merchant has spices, worth 25, 31, 40, 42, 45, 50 and 70c. per lb.; how many pounds of each may he take to form a compound of 300lb. at 37 c. per lb.?

3. A merchant mixes water with wines, worth 75, 90, 100 and 124c. per gal. so as to make a mixture of 3000gal. worth $1.08 per gal.; how many gal. of each may he take?

4. A drover has sheep, worth 9, 10, 15, 18 and 24s. each; how many of each may he take to form a flock of 160 sheep worth 16s. each ?

5. How many ounces of gold that is 18, 20, 23 and 24 carats fine, may be taken to form a mass of 30 ounces, that shall be 21 carats fine?

§ 34. SINGLE POSITION.

295. SINGLE POSITION is a method of solving an analytical question by assuming a number and working with it as though it were the true answer to the question.

296. RULE. Assume any number and proceed with it according to the conditions of the question; then say, as the result obtained is to the result given in the question, so is the assumed number to the required number.

Ex. 1. What number is that, which, being increased by and of itself, the sum will be 44?

on 6 to obtain 11, that

we have performed the same operations

we should have performed on 24 to obtain 44.. the proportion, 11 : 44 :: 6: 4th term, must give the number sought.

All examples in Single Position may be very easily analyzed ; thus in Ex. 1, the number sought is of itself;

, i. e. 44 is

the number is

and is ; now +3+3 of the number. If 44 is of the number, then of 44 is; of 44 is 4, and if 4 is, then g, or the whole number, is 6 times 4= 24, Ans. as before.

Let the learner solve the following examples both by Analysis and Position :

:

2. Divide 540 into 3 such parts that of the first, of the second and of the third shall be equal to each other.

The smaller number is of itself, and, by the nature of the question, the 2d number is of the 1st and the 3d is of the 1st; 540 is +3 += § of the 1st; if 540 is, then of 540 is; of 540 is 60, and if 60 is of the 1st number, then 2, or the whole, is twice 60 = 120, 1st number.

3. A teacher, being asked how many scholars he had, replied, if I had as many more, and as many more and 36 scholars, I should have 300; how many had he? Ans. 93.

4. A and B have the same income; A saves of his, but B, by spending twice as much as A, at the end of 4 years, finds himself $480 in debt; what is the annual income of each?

5. A man, being asked his age, replied, if have lived be multiplied by 9 and 3 of them be

Ans. $360.

of the years I subtracted from

the product, the remainder will be 150. How old was he?

6. Seven eighths of a certain number exceed of the same by 81; what is the number? Ans. 120. 7. A man lent a sum of money at 6 per cent., compound interest, and at the end of 3 years received the amount, $11910.16; what was the interest?

8. A gentleman bought a chaise, horse and harness for $470; the horse cost as much as the harness and the chaise as much as the horse; what was the price of each?

§ 35. DOUBLE POSITION.

297. DOUBLE POSITION is a method of solving an analytical question by assuming two numbers and working with each as though it were the true answer to the question.

298. RULE.—Assume any two numbers and proceed with each as the conditions of the question require; compare each result with the result given in the question and call each difference an error; multiply the 1st assumed number by the 2d error and the 2d assumed number by the 1st error; then, if both assumed numbers are too great or both too small, divide the difference of the products by the difference of the errors; but, if one assumed number is too great and the other too small, divide the sum of the products by the sum of the errors; in either case, the quotient will be the number sought.

Ex. 1. A gentleman, having a sum of money, spent $100 more than of it and had remaining $35 more than of it; how much had he at first?

$1000 × 315 = $315000, i. e. 1st assumed No. X 2d error. $1500 × 165 = $247500, i. e. 2d assumed No. X 1st error.

$67500 150 = $450, Ans.;

i. e. the difference of the products divided by the difference of the errors gives $450, the answer.

REMARK. This rule is applicable to all examples that can be solved by Single Position, and also to very many problems usually solved by Algebra. It is founded on the supposition that the 1st error is to the 2d error as the difference between the true and 1st supposed number is to the difference between the true and 2d supposed number. When this proportion does not hold, the problem cannot be solved directly by the rule.*

NOTE-Let the pupil solve the following examples, both by Position and Analysis.

* ALGEBRAIC DEMONSTRATION OF THE RULE. Having assumed the numbers a and b, and performed on them the operations required by the conditions of the example, let the results be represented by A and B, whereas, if we had assumed the true number, x, we should have obtained N, the result given in the example.

و با

Had r AND s both been negative, the value of x would not have been changed. Had r OR s been negative, the proportion would have taken one of two following forms, -rs: x -α: x -b, or r: -s :: x — a : x — rb + sa either of which, reduced, will give x = r+s agree with the enunciation of the rule.

; and these values of a

2. A and B have the same income; A saves of his, but B, by spending £30 per annum more than A, at the end of 8 years finds himself £40 in debt. What is their annual income?

Ans. £200 each.

3. A wine dealer bought 2 casks of porter, one of which held 3 times as much as the other; from each of these he drew 4gal., when 4 times as many gal. remained in the one as in the other. Required the number of gal. in each. Ans. 12 and 36.`

4. A and B have the same income; A saves of his, but B, by spending $200 per annum more than A, finds himself in debt. At the end of 5 years, A lends to B enough to pay his debt and has $250 left. What is the annual income of each?

5. What number is that which, being divided by 7, and the quotient diminished by 10, 3 times the remainder shall be 24? Ans. 126.

6. There is a fish whose head weighs 14 pounds, his tail weighs as much as his head and as much as his body, and his body weighs as much as his head and tail. What is the weight of the fish? Ans. 80lb.

7. A man hired a laborer for 50 days, on condition that for every day he worked he should receive $1.50 and for every day he was absent he should forfeit $1.75. At the expiration of the time he received $42.50. How many days was he absent?

Ans. 10.

8. A drover bought a number of horses, oxen and cows for $2640. For every horse he paid $50, for each ox as much as for a horse, and for each cow as much as for a horse. There were 3 times as many oxen as horses, and twice as many cows as How many were there of each?

oxen.

9. A gentleman has 2 horses and a saddle. The saddle is worth as much as the 1st horse, and if it be put on the 1st horse, they together will be worth .3 as much as the 2d horse. If the saddle be put on the 2d horse they will be worth 3 times as much as the 1st horse. What is the value of the 2d horse?

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