COMPOUND SUBTRACTION TEACHES to find the difference, inequality, or excess, between any two sums of divers denominations. RULE.* Place those numbers under each other, which are of the same denomination, the less being below the greater; begin with the least denomination, and, if it exceed the figure over it, add as many units as make one of the next greater; subtract it therefrom; and to the difference add the upper figare, remembering, always, to add one to the next superior denomination, for that which you added before. * The reafon of this Rule will readily appear, from what was faid in Simple Subtraction; for the adding depends upon the fame principle, and is only dif ferent, as the numbers to be fubtracted are of different denominations. = In the 1st Ex. of English Money, I take 1 qr. from 1 qr. and set down 0, the remainder. Because I cannot take eight from 6 pence, I add to 6, 12 pence which make a shilling, and from 18 take 9, and set down 10, the difference. As I added 12 pence 1 shilling to the upper pence, I now carry 1 shilling to the lower shillings, and take 12 from 15, and set down 3, the remainder. The rest of the process is evident. It is obvious that a similar course must be pursued in the Examples under the several weights and measures. 3. TROY WEIGHT. PROBLEMS RESULTING FROM A COMPARISON OF THE PRECEDING RULES. PROB. 1. Having the sum of two numbers, and one of them given, to find the other. Rule. Subtract the given number from the given sum, and the remainder will be the number required. Let 288 be the sum of two numbers; one of which is 115, the other is required? From 288 the Sum, Take 115 the given number. PROB. 2. Having the greater of two numbers, and the difference between that and the less given, to find the less. Rule. Subtract the one from the other. Let the greater number be 325, and the difference between that and the other, 198: What is the other? From 325 the greater, PROB. 3. Having the least of two numbers given, and the dif ference between that and a greater, to find the greater. Rule. Add them together. Sum 325 the greater number required. PROB. 4. Having the sum and difference of two numbers given, to find those numbers. Rule. To half the sum add half the difference, and the sum is the greater, and from half the sum take half the difference, and the remainder is the less. Or, from the sum take the difference, and half the remainder is the least: to the least add the given difference, and the sum is the greatest. What are those two numbers, whose sum is 48, and difference 14? 2)48 2)14 24+7-31 the greater, and 24-7-17 the less. sum=24 diff.=7 Or 48-14-2=17, & 17+14=31. This rule is obvious on considering any example in the following manner. Thus, let the two numbers be 32 and 46; their sum is 46+32, and their difference is 46-32. Now 46+32÷2+46-32 ÷2=46+32+46—32÷÷2=46+46-246, the greater. And 46+32÷÷2¬46+32÷÷2÷46+32—46+32÷÷2=32+32+2=32, the less. For, in the first case, the 32 to be subtracted from 32, leaves nothing; and, in the latter, the 46 is balanced by the other 46, and you have only 32+32—2=32. The preceding and following problems are evident from the rules of Addition and Subtraction, Multiplication and Division. PROB. 5. Having the product of two numbers, and one of them given, to find the other. Rule. Divide the product by the given number, and the quotient will be the number required. |