48. SUBTRACTION is taking a less number from a greater number of the same kind, to find their difference. The greater number is called the MINUEND; the less number is called the SUBTRAHEND; and the result is called the DifferENCE or REMAINDER. Ex. 1. Arthur had 7 apples, but he has given 4 of them to Mary; how many apples has he now? Ans. 3; because 4 apples taken from 7 apples leave 3 apples. 2. John having 17 marbles, lost 7 of them; how many had he left? 49. The sign of subtraction,—, called minus, signifies that the number after it is to be taken from the number before it; thus, 7 —4—3, i. e. seven minus four, or, seven diminished by four, equals three. 3. How many are 10 - 6? 4. How many are 12-8? 12-4? 16-6? Ans. 4. NOTE. When the numbers are small, the subtraction is readily performed in the mind; but when they are large, the work is more easily done by writ ing the figures, as in the following examples. 50. To subtract when no figure in the subtrahend is greater than the corresponding figure in the minuend. 5. From 796 take 582. 48. What is Subtraction? Minuend? Subtrahend? Remainder? 49. Make the sign of subtraction. Its meaning? How do we subtract when the numbers are small? How when they are large? 14. A farmer bought a farm for $4875 and sold it again for $3463; how much did he lose by the transactions? Ans. $1412. 15. By the census of 1860, the population of Maine was 628276, and that of New Hampshire was 326072; how many more people were there in Maine than in New Hampshire? 16. If I borrow $4687 and afterwards pay $2423, how much do I still owe? 51. To subtract when any figure in the minuend is less than the corresponding figure in the subtrahend. 17. From 483 take 257. OPERATION. Minuend, 483 Remainder, 257 There are two methods of explaining this operation: 1st. As we cannot take 7 units from 3 units, one of the 8 tens is put with the 3 units, making 13 units, and then, 7 units from 13 units leave 6 units. Now as one of the 8 tens has been put with the 3 units, only 7 tens remain in the minuend, and 5 tens from 7 tens leave two tens, and, finally, 2 hundreds from 4 hundreds leave 2 hundreds; .. the entire remainder is 226. 2d. Instead of taking away 1 of the 8 tens in the minuend, we may add 1 ten to the 5 tens in the subtrahend, and then take the sum (6 tens) from the 8 tens, since the result is 2 tens by either process. The second mode depends on the principle, that, if two numbers are equally increased, the difference between them remains unchanged; thus, the difference between 9 and 4 is 5, and, if 10 is added to both 9 and 4, making 19 and 14, the difference still is 5. Now, in solving Ex. 17 by the second method, we add 10 units to the minuend and 1 ten (the same as 10 units) to the subtrahend, and .. find the same remainder as by the first method. 51. How many methods of subtracting when a figure of the minuend is less than the one under it? What is the first method? Second? The second depends on what principle? By the second method, is the same number added to minuend and subtrahend? How? 52. The preceding examples illustrate all the principles in subtraction. Hence, to perform subtraction, RULE. 1. Write the less number under the greater, units, under units, tens under tens, etc., and draw a line beneath. 2. Beginning at the right hand, take each figure of the subtrahend from the figure above it, and set the remainder under the line. 3. If any figure in the subtrahend is greater than the figure above it, add TEN to the upper figure and take the lower figure from the SUM; set down the remainder and, considering the next figure in the minuend ONE LESS, or the next figure in the subtrahend ONE GREATER, proceed as before. 53. PROOF. Add the subtrahend and the remainder to gether, and the sum should be the minuend. NOTE 1. This proof rests upon the self-evident truth, that the whole of a thing is equal to the sum of all its parts; thus, the minuend is separated into the two parts, subtrahend and remainder; hence the sum of those parts must be the minuend. 14. A farmer bought a farm for $4875 and sold it again for $3463; how much did he lose by the transactions? Ans. $1412. 15. By the census of 1860, the population of Maine was 628276, and that of New Hampshire was 326072; how many more people were there in Maine than in New Hampshire? 16. If I borrow $4687 and afterwards pay $2423, how much do I still owe? 51. To subtract when any figure in the minuend is less than the corresponding figure in the subtrahend. 17. From 483 take 257. OPERATION. Minuend, Subtrahend, Remainder, 483 257 There are two methods of explaining this operation: 1st. As we cannot take 7 units from 3 units, one of the 8 tens is put with the 3 units, making 13 units, and then, 7 units from 13 units leave 6 units. Now as one of the 8 tens has been put with the 3 units, only 7 tens remain in the minuend, and 5 tens from 7 tens leave two tens, and, finally, 2 hundreds from 4 hundreds leave 2 hundreds; .. the entire remainder is 226. 2d. Instead of taking away 1 of the 8 tens in the minuend, we may add 1 ten to the 5 tens in the subtrahend, and then take the sum (6 tens) from the 8 tens, since the result is 2 tens by either process. The second mode depends on the principle, that, if two num bers are equally increased, the difference between them remains unchanged; thus, the difference between 9 and 4 is 5, and, if 10 is added to both 9 and 4, making 19 and 14, the difference still is 5. Now, in solving Ex. 17 by the second method, we add 10 units to the minuend and 1 ten (the same as 10 units) to the subtrahend, and .. find the same remainder as by the first method. 51. How many methods of subtracting when a figure of the minuend is less than the one under it? What is the first method? Second? The second depends on what principle? By the second method, is the same number added to minuend and subtrahend? How? 52. The preceding examples illustrate all the principles in subtraction. Hence, to perform subtraction, RULE. 1. Write the less number under the greater, units, under units, tens under tens, etc., and draw a line beneath. 2. Beginning at the right hand, take each figure of the subtrahend from the figure above it, and set the remainder under the line. 3. If any figure in the subtrahend is greater than the figure above it, add TEN to the upper figure and take the lower figure from the SUM; set down the remainder and, considering the next figure in the minuend ONE LESS, or the next figure in the subtrahend ONE GREATER, proceed as before. 53. PROOF. Add the subtrahend and the remainder to gether, and the sum should be the minuend. NOTE 1. This proof rests upon the self-evident truth, that the whole of a thing is equal to the sum of all its parts; thus, the minuend is separated into the two parts, subtrahend and remainder; hence the sum of those parts must be the minuend. |