Then 15 cancels 15, and 3 cancels 9, and gives 3. Lastly, 4.3.10=120, the number of masons required. 2. If, when money is loaned at 6 per cent. per annum, A borrows of B £26 for 120 days, how long, in order to cancel the obligation, must A lend B £17 6 s. 8 d., loaned at 41 per cent.? is when money Áns. 240 days. 3. If, when money is at 5 per cent., B borrows of A $840 for 8 mos., how long must A retain $636,36,4 of B's money, to cancel the obligation, when money is at 7 per cent. ? Ans. 7 mos. 1. d. 4. If 7 horses eat 231 bushels of oats in 132 days, how many days will 561 bush. serve 11 horses, at the same rate? Ans. 204 days. 5. If a troop of horse march 506 miles in 23 days of 10 hours, in how many days of 8 hrs. will it march 704 miles, at the same rate? Ans. 40 days. 6. If two men mow 9 acres of grass in 3 days, working 8 hrs. per day, how many men will mow 27 acres in 4 days, working 9 hrs. per day? Ans. 4 men. 7. If a garrison of 1200 men is victualled for 6 mos. of 30 days, so as to allow each man 2 lbs. weight per day, how much must each man be allowed per day to make the provisions last 8 mos., supposing it reinforced with 300 men, bringing with them a supply of 108000 lbs. ? Ans. 1 lbs. 8. A garrison of 1000 men, victualled for 8 mos. of 30 days, at the daily rate of 21 lbs. per man, is besieged 40 days; but, making a sortie, they drive off the enemy; now suppose that, having lost 200 men, they are immediately reinforced with 400, who bring 40000 lbs. of provisions, what must each man's daily ration be to make the provisions last? Ans. 21 lbs. SECTION XXII. POSITION-ALLIGATION-PERMUTATION COMBINATION. Position. ARRANGEMENT 433. POSITION, or SUPPOSITION, is that branch of arithmetic which finds a number sought from a number, called the supposition, which is arbitrarily assumed, and is operated upon, ac cording to the conditions of the question, exactly as though it was the true number. Position is called Single or Double, accordingly as the result is obtained from one or two suppositions. Those questions the data of which require only that the quantity operated upon shall be increased or diminished in a certain ratio or by quantities which are aliquot parts or multiples of itself, belong to Single Position, and may be solved by the following geometrical proportion : As the result of the operation performed on the supposition, To the true result, or given number, So is the supposed number To the number sought. This rule may be thus elucidated: Let A be the supposed and B the true number, or number sought. Then is the ratio of A to B. But A is operated upon, according to the conditions of the question, by multiplication or division, or both, in the same manner as B, and therefore both are increased or both diminished in the same ratio. Let q represent this qA A ratio; then (167) because qB B' : qB:: A: B, and hence the rule. Examples. it is plain (402) that qA 1. A and B trade with equal capital: A gains 25 per cent. ; B loses 10 per cent., after which A has $5000 more than B. What sum had each at first? Suppose $10000: Then, as 25 p. c. = 4, and 10 p. c. = To: 10000 + 10000 4 10000 10000 10 10000+2500 10000-1000 = 12500 incr'd capital of A. 9000 dim'd capital of B. 3500 diff. which should be 5000. 3500 5000 Then by the Rule, 10000, or 7:10:: 10000: 14285,71%, Answer. 2. A man bequeaths $5000 to A, B, and C, so that A's part may be double that of B, and the share of B to that of C as 4 to 5. What is the share of each? Ans. A, $2352,94,2; B, $1176,471; C, $1470,5814. 3. A miller wishes to make a quantity of feed by grinding 400 bush. of barley with peas and beans, so that the peas and beans together may be to the barley as 7 to 3, and the peas double of the beans. How many bushels of each must he grind ? Ans. 311 bu. beans, and 6223 bu. peas. 4. Says A to B, My horse without the saddle is worth your horse with my saddle on his back. True, replies B, but my saddle alone is worth 4 times as much as your horse, 3 times as much as your horse and saddle, and, together with the value of my horse, is worth $1000. What is the value of each horse and saddle? 1 Ans. A's horse, $240; B's horse, $160; A's saddle, $80; B's, $960. Double Position Absolute. 434. Those questions, from the data of which the quantity operated upon must be increased or diminished, not in a certain ratio, require two suppositions, with each of which we find the result as in Single Position. If either of these results accords with the given number, the supposition which gave it must be the number sought. But if both differ, find the difference between each result and the given number, and call this difference an error in plus, when the result is greater; or an error in minus, when the result is less than the given number. Next write the two suppositions under each other, and opposite each the error which it gave, preceded by its appropriate sign, plus or minus. Multiply the first supposition by the error of the second, and the second supposition by the error of the first, in a cross order. Then, if the errors have like signs, that is, both plus or both minus, divide the difference of the products by the difference of the errors. But, if they have unlike signs, that is, the one plus and the other minus, divide the sum of the products by the sum of the errors. The quotient is, in either case, the number sought. Now when S' is greater than S, it is evident that e' will be plus. Also that the greater S' is, the greater will e' be. The errors, therefore, when both suppositions are greater, are as the excesses of the suppositions; that is, S'S: S"-S:::e"; hence (403) = S"e-Se' S'e"- Se", and by transposition (398) = S(e" — e') = S'e"- S'e, and consequetly S S'e"-S"e It is evident (408) that by inverting the terms of each ratio in the above proportion, or (398) by changing the sign of each term in the equation resulting from it, we shall obtain S"e-S'e" S= Hence it is of no consequence, as to the rule, which product is the greater. Again, when S' is less than S, e' will be minus, and the less S' is, the greater will e' be. The errors, therefore, when both suppositions are less, will be to each other in direct ratio, as the diminutions of the suppositions; that is, S—S': S— S" : : e' : e", and hence S'e"S"e = e" S (e" —e') = S'e" -- S"e', or finally S That is to say, when the errors are both plus, or both minus, the difference of the products, divided by the difference of the errors, gives the number sought. Lastly, for the case in which the signs of the errors are unlike: Let S" be less than S, and consequently e' minus; and let S" be greater than S, and consequently e" plus. Then the greater the difference between S' and S, the greater will e be; also, the greater the difference between S" and S, the greater will e" be; therefore, S-S': S" - S: : é: e"; and putting this in equation, or S (e" + e') = S"e' + S'e", or finally S = that is to say, when one error is plus and the other minus, the sum of the products, divided by the sum of the errors, gives the number sought. Double Position Approximate. 435. We must here inform the student that the above is no longer true when the data of the question require the involution of S, or any part of it, into itself; for, in this case, instead of finding, from the formula, the true result, we only arrive at an approximation. Put 6:10 S'S: S" - See" 8; Now suppose the numbers S', S", and S to have been involved, so that instead of each number, we had its square. Then, that the proportion may remain, it will be necessary to show that the differences of the squares of the numbers are as the differences of the numbers themselves. Let S' S" 10, and S-6. Then the above proportion becomes 8. 6 ee", or 2: 4 :: ee". Then instead of the numbers, taking their squares, if the proportion still exists, we shall have 64 36 100 -36 :': 2:4; that is, 28 : 64 : : 2 : 4, or (403) 112 128, which is absurd. We shall, however, obtain an approximation. For, let a and a+q be the suppositions, both greater than c, the true number. Also put a qx; then a-c: x — c::e': e", and Then taking the square of each ac ratio of differences. х с = supposition and the square of the true number, we have ratio of the involved numbers. To show the ap a2 proximation of these ratios, we divide one by the other, thus: Now, if we take each supposition as near as possible to the true number, their difference q is a very small number, and consequently, 1+ a near approximation to a unit, which a + c is the result when the ratios are equal. To solve a question of this kind, therefore, assume two numbers, each as near as possible to the true, and find the result as above, by the appropriate formula. Take the result which is an approximation, and of the two suppositions, that which is nearest to the true number, or any other which may appear still |