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-88 Position.

5. A Goldsmith sold 1 lb. of gold, at 2 cts, for the first ounce, 8 cents for the second, 32 cents for the third, &c.in a quadruple proportion geometrically: what did the whole come to ? Ans. $11 IS-18, 10 cos. 6. What debt can be discharged in a year, by paying 1 farthing the first month, 10 farthings, or (2}d) the second and so on, each month in a tenfold proportion? Ans. E115740740-14s 9d. 3 qrs. 7. A thrasher worked 20 days for a farmer, and received for the first days work four barley-corns, for the second 12 barley corns, for the third 36 barley corns, and so on, in triple proportion geometrically. I demand what the 20 day's labour came to supposing a pint of barley to contain 7680 corns, and the whole quantity to be sold at 2s. 6d. per bushelf Ans. E1773 7s 6d. rejecting remainders 8. A man bought a horse, and by agreement, was to ive a farthing for the first nail, two for the second, foul É. the third, &c. There were four shoes, and eight nails in each shoe; what did the horse come to at that rate 7 Ans. E44739:24 5s 3d 9. Suppose a certain body, put in motion, should move the length of 1 barley-corn the first second of time, one inch the second, and three inches the third -cond of time, and so continue to increase its motion in triple proportion geometrical; how many yards would the said body move in the term of half a minute. Ans. 953199.685623 yds, 1 ft. 1 in. Ib, which is no less than five hundred and forty-one millions of miles.

- - - POSITION.

POSITION is a rule which, by false or supposed num bers, taken at pleasure, discovers the true ones required.It is divided into two parts, Single or Double.

SINGLE, I-OSITION

IS when one number is required, the properties of which are given in the question.

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Rule.—1. Take any number and perform one same operation with it, as is described to be performed in the question. Then say; as the result of the operation : is to the given sum in the question : : so is the supposed number: to the true one required. The method of proof is by substituting the answer in the ques tion.

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1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I now have, half as many, one-third, and one fourth as many, I should ther bave 148; How many scholars had he Suppose he had 12. As 37 : 148 : : 12 : 48 Ans.

as many = 12 48 as many = 8 24 # as many = 4 16 : as many – 3 12 Result, 37 Proof, 148 2. What number is that which being increased by 4, , and of itself, the sum will be 125? Ans. 60.

3. Divide 93 dollars between A, B and C, so that B's share may be half as much as A's, and C's share three times as much as B's. Ans. A's share $31, B's $154, and C's $464. 4. A, B and C, joined their stock and gained 360 dols. of which A took up a certain sum, B took 3 times as much as A, and C took up as much as A and B both; what share of the gain had each? Ans. A $40, 13 $140, and C $180. 5. Delivered to a banker a certain sum of money, to receive interest for the same at til, per cent per annum, simple interest, and at the end of twelve years received 7311. principal and interest together; what was the sum delivered to him at first 7 - Ans. E425. 6. A vessel has 3 cocks, A, B and C : A can fill it in 1 hour, B in 2 hours, and C in 4 hours; in what time will they all fill it together? Ans 34 min. 174 sec.

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90 Douei.e. Positrox.

DOUBLE POSITION,

TEACHES to resolve questions by making two suppo sitions of false numbers.”

IRULL.

1. Take any two convenient numbers, and proceed with each according to the conditions of he question. 2. Find how much the results are different from the re. sults in the question. 3. Multiply the first position by the last error, and the las: position by the first error. 4. If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer. 5. If the errors are unlike, divide the sum of the products by the sum of the errors, and the quotient will be the answer. Note:-The errors are said to be alike when they are both too great, or both too small; and unlike, when one is too great, and the other too small.

Ex-Mi-Es.
1. A purse of 100 dollars is to be divided among 4 men
A, B, C and D, so that B may have four dollars more that
A, and C S dollars more than B, and D twice as many as
C; what is each one's share of the money?

1st. Suppose A 6 2d. Suppose A 8 B 10 B 12 C 18 C 20 * * * * * 70 So 100 100 1st error, so 2d error, 20

* Those questions in which the results are not proportional to their post tions, belong to this rule; such as those in which the number sought is it. . creased or diminished by some given number, which is no known part of to | number required. /

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10)120(12 A's part. 2. A, B, and C, built a house which cost 500 dollars, of which A paid a certain sum : B paid 10 dollars more than A, and C paid as much as A and B both ; how much did each man pay ons. -1 paid $120, BS130, and C$250. 3. A man bequeathed 1001 to three of his friends, after this manner; the first must have a certain portion, the second must have twice as much as the first, wanting 81 and the third must have three times as much as the first, wanting 151. I demand how much each man must have? ons. The first £20 10s. second 533, third, £46 10s. 4. A labourer was hired for 60 days upon this condition; that for every day he wrought he should receive 4s. and for every day he was idle should forfeit 2s. ; at the expiration of the time he received 71 10s. ; how many days did he work, and how many was he idle? ins. He wrought 45 days, and was idle 15 days. 5. What number is that which being increased by its 1, ls o and 18 more, will be doubled 2 1ns, 72. A man gave to his three sons all his estate in money, viz. to F hals, wanting 501 to G one-third, and to H the test, which was 101. less than the share of G: I demand one sum given, and each man's part 1

ons, the sum given was £360, whereof F had £130,

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102 PERMUTATION OF QUANTITLEs.

7. Two men, A and B, lay out equal sums of money in trade: A gains 1261 and B loses 871 and A's money is now double to B's; what did each lay out? - Ans. E300. 8. A farmer having driven his cattle to market, received for them all 1301, being paid for every ox 71 for every cow 5l. and for every calf # 10s, there were twice as many cows as oxen, and three times as many calves as cows: how many were there of each sort? Ans. 5 ozon, 10 cows, and 30 calves. 9. A, B, and C, playing at cards, staked 324 crowns: but disputing about tricks, each man took as many as he could; A got a certain number; B as many as A and 15 more; C got a 5th part of both their sums added together; how many did each get? Ana. A got 1274, B142), C 54.

PERMUTATION OF QUANTITIES,

IS the showing how many different ways any given number of things may be changed.

To find the number of Permutations, or changes, that can be made of any given number of things all different from each other.

Rule-Multiply all the terms of the natural series of number

from one up to the given number, continually together, and the oxproduct will be the answer required

Ex-MPLEs.

1. How many changes can be 1 a b c made of the first three letters of 2 a c b the alphabet? Proof, o . 5 c h a 1 x2 x3–6. Ans. to cab

2. How many changes may be rung on 9 bells?
Ans, 362880.

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