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26. Addition of United States Money. In adding United States money simply proceed as with ordinary integers, $12.50
keeping units of the same order in the same 0.75
column as here shown. 23. We can easily keep the units of the same order in the
1.05 same column by placing the decimal points under one $37.30
SUBTRACTION OF INTEGERS
27. Subtraction. The process of finding what number must be added to one of two numbers to make the other, or of taking one number from another, is called subtraction.
28. Symbol. The symbol of subtraction is a short horizontal line, –. It is read minus, a word meaning less.
Thus 17 – 9 = 8 is read "17 minus 9 equals 8."
29. Terms. The smaller of the two given numbers in subtraction is called the subtrahend; the larger one is called the minuend; the result of the subtraction is called the remainder or difference. To check the result, add the subtrahend and remainder; the result should be the minuend.
30. The Addition Method of Subtraction. This method, also called the Austrian or making-change
468 minuend method, is recommended in the New York
143 subtrahend State Syllabus. It consists simply in find
325 remainder ing, in the case of 468 – 143, what must be added to 143 to make 468. We proceed as follows: 3+5=8; 4+2=6; 1+3 = 4.
31. The Taking-away Method. This is, at present, a more familiar method of subtraction. Here we say : 3 from 8,5; 4 from 6, 2; 1 from 4, 3.
Of the two methods the first is always used in making change, it is the more rapid when thoroughly learned, and it is growing in favor. When a pupil has already acquired one method, however, and works rapidly and accurately, it is not good policy to attempt to change it.
41. A grocer had 1360 pounds of sugar and sold 1040 pounds. How many pounds had he left?
42. A farmer had 2445 bushels of corn and sold 2120 bushels. How many bushels had he left ?
43. A man's income for a year was $1975 and his expenses were $1630. How much had he left?
44. A dealer had 2796 tons of coal and sold 1385 tons. How many
tons had he left ? 45. A dry goods merchant had 1440 spools of thread and sold 1200 spools. How many spools had he left ?
46. A city school had 1296 pupils of whom 184 were in the first-year class. How many were in all the other classes together?
47. A real estate dealer had 1278 acres of land for sale, and sold 1163 acres. How many acres had he left unsold ?
48. The distance from New York to Albany is 143 miles, and from New York to Chicago is 975 miles. How far is it from Albany to Chicago by this route?
49. The area of Indiana is 36,350 square miles, and that of Rhode Island 1250 square miles. What is the difference in area ?
50. The area of Alabama is 52,250 square miles and that of California is 158,360 square miles. What is the difference in area ?
51. The area of Nebraska is 77,510 square miles, and that of South Dakota is 77,650 square miles. What is the difference in area ?
52. In a recent year Pennsylvania produced 31,816,496 bushels of oats, and Massachusetts 214,472 bushels. What was the difference in the amounts ?
32. General Case of Subtraction. Suppose we wish to find the difference between 652 and 476. Here we may use either of the methods stated in $ 30 and $ 31.
(1) The addition (Austrian) method uses this principle : If the same number is added to both minuend and subtrahend, the remainder is not changed. Thus 8 – 2 gives the same result as 18 – 12 or 108 — 102.
Taking the case of 652 476, we now consider what number added to 6 makes 2. Since it is impossible to find such a number, what number added to 6 makes 12 ? Evidently 6, which we write below. But we increased the 652 minuend minuend from 2 to 12 by adding 10, so we add 476 subtrahend 10 to the subtrahend, making the 7 into an 8. 176 remainder Then, since no number added to 8 makes 5, what number added to 8 makes 15 ? Evidently 7, which we write below. But we increased the minuend from 5 tens to 15 tens, by adding 100 (or 10 tens), so we add 100 to the subtrahend, making the 4 a 5. Then 5 + 1 = 6, and we write the 1 below.
(2) The taking-away method. Considering the same problem, we proceed as follows: Since we cannot take 6 from 2 we change one of the 5 tens to units, and this with the 2 units makes 12 units; then 12 – 6 = 6. Since we have 500 + 140 + 12 used one of the tens in the 5 tens we have 4 tens 400 + 70 + 6 left, from which we cannot take 7 tens. We 100 + 70 + 6 change one of the 6 hundreds to tens, and this with the 4 tens makes 14 tens; then 14 tens
7 tens. We now have 5 hundreds left of the 6 hundreds, and 5 hundreds – 4 hundreds = 1 hundred. The remainder is therefore 176. It will make the work clearer to see the numbers separated as here shown, but this is not done in practice.
33. Special Cases. In a case like 5000 — 2765 we may shorten the work by thinking of the 5000 as 4990 + 10, and of 2765 as 2760 + 5, beginning to subtract at the left.
This amounts to taking the 2 from 4 and 4 9 9 10 every other figure from 9 except the units, which 2 7 6 5 we take from 10. The remainder is 2235.
2 2 3 5 In subtracting United States money place the decimal points under one another and proceed as with integral numbers.