DOUBLE POSITION. Rule. 1 st. Assume two numbers or quantities, and work with them according to the conditions of the calculation. 2 ndly. If neither of the results answers to the given result, find the error of each, and multiply each assumed number by its opposite error. 3 rdly. If the errors are both greater or both less than the given number, take their difference for a divisor, and take the difference of their products for a dividend; but if one error is greater and the other is less than the given number, take their sum for a divisor, and the sum of their products for a dividend, and in either case the quotient will be the number sought. EXAMPLE. To divide £ 200 between three persons, A, B, and C, so that A may have £ 6 more than B, and B £ 8 more than C. Ex. 1. The ages of 3 persons amount to 180 years; the first is 6 years older than the second, and the third is 24 years younger than the second; what is the age of each? Answer 72, 66, and 42 years. Ex. 2. What are those unequal weights, to the less of which, if you add 5 lb. the amount will be double the greater weight, but if the same be added to the greater, the amount will be triple the less weight. Answer 3 lb. and 4 lb. Ex. 3. What is the amount of that sum of money from which if 1⁄2, 1, and of it be taken, the remainder will be £ 25? 'Answer £ 100. N. B. For many of the purposes to which Double Position is applied, the following Rule is more convenient than the above. PERMUTATIONS. Permutation is the varying of the position of the different given things, without varying the number of the whole. Rule. Multiply the numbers of the position of the given terms into one another. Example. To find how many permutations or changes can be made with 6 things? 1 X 2 X 3 X4 X5 X6 720 Answer. EXERCISES. Ex. 1. How many days can 5 persons be placed in a different position round a table at dinner? Answer 120 days. Ex. 2. How many changes can be rung upon 10 Bells, and how long would they be ringing one over, supposing 10 changes to be rung per minute? Answer, Changes 3,628,800 362,880 252 days. COMBINATIONS. Combination is the taking of a less number of things out of a greater, and combining them together without regard to their order. Rule. Take for a divisor the combined product of the series 1, 2, 3, 4, &c., up to the less number to be taken each time; then take for a dividend the continued product of a series of the same number of times, decreasing by 1 from the greatest number of times out of which the combination is to be made, and find the quotient of the latter product divided by the former, for the number of the combinations required. Find by trial two numbers as near the true number as convenient; use each in the place of the one required, and find each result. Then say, by proportion, as the difference of these results is to the difference of the two assumed numbers, so is the difference between the true result given and either of the former, to a correction which is to be applied, in addition or subtraction, to the assumed number whose result is used. Thus, with the example, the difference of the results being 60, the difference of the assumed numbers being 20, and the difference between the true result (200) and the first result (160) being 40, we say, as 60 is to 20 so is 40 to 13 and one third, which, added to 60, gives A's share. EXAMPLE. To find how many combinations of 4 at a time, can be made out of 24 Letters of the Alphabet ? 1 X 2 X 3 X 4 = 24 24 X 23 X 22 X 21 = 255024 24 ) 255024 10626 Number of combinations. EXERCISES. Ex. 1. How many combinations can be made of 6 letters out of 12? Answer 924 combinations. Ex. 2. In how many ways can 4 men be taken for sentinels out of a company of 60 men? Answer 487635 ways. COMPOSITIONS. Composition is the taking of a given number of things out of as many equal rows or sets of different things, one out of each row, and combining them together. Rule. Multiply the whole number in each set as many times in succession together as there are sets of things. EXAMPLE. To find how many compositions can be made in 4 sets of 9 things each, so as to have 4 each time, viz. 1 out of each set. 9 X 9 X 9 X 96561 EXERCISES. Answer. Ex. 1. In 6 sets of things of 10 in each, how many compositions can be made taking 6 each time, viz. 1 out of each set? Answer 1000000. Ex. 2. How many compositions can be made with the numbers on 6 dice of 6 faces each. Answer 46656. THE END. A continuation of this work has been published under the following title: AN APPENDIX TO THE COMMERCIAL ARITHMETIC, EXHIBITING THE METHODS EMPLOYED BY MERCHANTS, BANKERS, AND BROKERS, FOR THE VALUATIONS OF MERCHANDISE, MENTAL PER CENTAGES, INTEREST ACCOUNTS IN ACCOUNTS-CURRENT, THE PUBLIC FUNDS, MARINE INSURANCES, THE STANDARDING OF GOLD AND SILVER, THE ARBITRATIONS OF EXCHANGE IN BILLS, BULLION, AND MERCHANDISE, AND ACTUAL AND PRO-FORMA STATEMENTS OF BRITISH AND FOREIGN INVOICES AND ACCOUNT-SALES. A MANUAL OF FOREIGN EXCHANGES, in the direct, indirect, and cross operations of Bills of Exchange; including an extensive investigation of the Arbitrations of Exchange in Bills, Coin, and Bullion. Second Edition. By W. Tate. KEYS to the above. BUNGAY: PRINTED BY J. R. AND C. CHILDS. TO THE COMMERCIAL ARITHMETIC, EXHIBITING THE METHODS EMPLOYED BY MERCHANTS, BANKERS, AND BROKERS, FOR THE VALUATIONS OF MERCHANDISE, MENTAL PER CENTAGES, INTEREST ACCOUNTS IN ACCOUNTS-CURRENT, THE PUBLIC FUNDS, MARINE INSURANCES, THE STANDARDING OF GOLD AND SILVER, THE ARBITRATIONS OF EXCHANGE, IN BILLS, BULLION, AND MERCHANDISE, AND ACTUAL AND PRO-FORMA STATEMENTS OF BRITISH AND FOREIGN INVOICES AND ACCOUNT-SALES. By W. TATE, JUNR. SECOND EDITION. LONDON: EFFINGHAM WILSON, ROYAL EXCHANGE. W. F. WAKEMAN, DUBLIN. WAUGH AND INNES, EDINBURGH. MDCCCXXXVII. |