RULE. 1. Take any number and perform the same operation with it, as is described to be performed in the question. 2. 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 question. XAMPLES as many 1. A schoolmaster being asked how many scholars he nad, said, If I had as many more as I now have, half as many, one-third and one-fourth as many, I should then have 148 ; How many scholars had he? Suppose he had 12 As 37 : 148 : : 12 : 48 Ans, 12 48 24 16 das many 12 Result, 37 Proof, 148 2. What number is that which being increased by }, ý, and į of itsell, 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 $15, and C's $461, 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. w $40, B $140, and C $180. 5. Delivered to a banker a certain sum of money, to receive interest for the same at 61. 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? Ars. €425. 6. A vessel has 3 cocks, A, B and C ; Acan fill it in 1 hour, B in 2 hours, and C in 4 hours ; in what ime will they all fill it together? Ans. 34min, 17 sec. 202 DOUBLE POSITION TEACHES to resolve questions by making two suppositions of false numbers. * RULE. 1. Take any two convenient numbers, and proceed with each according to the conditions of the question. 2. Find how much the results are different from the results in the question. 3. Multiply the first position by the last error, and the last 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 quotieni 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. EXAMPLES. 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 than A, and C 8 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 A8 B 10 B 12 C: 18 C20 D 36 D 40 * Those questions in which the results are not proportional to their positions, belong to this rule ; such as those in which the number sought is increased or diminished by some given number, which is no known part of the number required. The errors being alike, are both too small, therefore, Pos. Err A 12 X 20 Proof 100 120 240 10) 120/12 A's part. 2. A, B, and C, built a house which cost 500 dollars, of which À 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? Ans. A paid $120, B $150, 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 8l. and the third must have three times as much as the first, wanting 15l. ; I demand how much each man must have ? Ans. The first £20 10s. second £33, 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 28.; at the expiration of the time he received 71. 10s.; how many days did he work, and how many was he idle ? Ans. He wrought 45 days, and was idle 15 days. 5. What number is that which being increased by its , its i, and 18 more, will be doubled ? Ins. 72. 6. A man gave to his three sons all his estate in money, viz. to F half , wanting 501. to G one-third, and to H the rest, which was 10l. less than the share of G; I demand he sum given, and each man's part ? Ans, the sum given was £360, whereof F had £130, G £120, and # $110. 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. £300. 8. A farmer having driven his cattle to market, receive ed for them all 1301. being paid for every ox 71. for every cow 51. and for every cali il. 10s. there were twice as many cows as oxen, and three times as many calves as cows ; ow many were there of each sort ? Ans. 5 0.xon, 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 gut 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? Ans. A got 127), B 1423, C 54. PERMUTATION OF QUANTITIES, Is the snowing 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 froni each other, RULE. Multiply all the terms of the natural series of numbers, from one up to the given number, continually together, and the last product will be the answer required. EXAMPLES. bac bca 1. How many changes can be abc made of the first turee letters of 2 ась the alphabet ? 3 Proof, 5 cba 1x2x36 fns. 6 са в 2. How many changes may be rung on 9 bells ? Áns, 362880 3. Seven gentlemen met at an inn, and were so welt pleased with their host, and with each other, that they agreed to tarry so long as they, together with their host, couid sit every day in a different position at dinner ; how long must they have staid at said inn to have fulfilled their agreement ? Ans. 11037 years. COMPUTED AT CASE I. Compound Interest. RULE. 1. Make 1 the first term of a geometrical progression, and the amount of $1 or £1 for one year, at the given rate per cent. the ratio. 2. Carry on the series up to as many terms as the given number of years, and find its sum. 3. Multiply the sum thus found, by the given annuity, and the product will be the amount sought. EXAMPLES. 1. If 125 dols. yearly rent, or annuity, be forborne, (or unpaid) 4 years ; what will it amount to at 6 per cent. per annum, compound interest ? 1+1,76+1,1236+1,191016=1,374616 sum of the series.* _ Then, 4,374616X125=$546,827 the amount sought. OR BY TABLE II. Multiply the Tabular number under the rate and opposite to the time, by the annuity, and the product will be the amount sought. * The sum of the series thus found, is the amount of il: or 1 dollar annuity, for the given time, which may be found in Table II. ready calculated. Hence, either the amount or present worth of annuitie's nay be readily found by tables for that purpose: $ |