47. The Difference or Remainder is the result obtained by the process of subtraction.

The minuend and subtrahend must be like numbers; thus, 5 dollars from 9 dol. lars leave 4 dollars; 5 apples from 9 apples, leave 4 apples; but it would be absurd to say 5 apples from 9 dollars, or 5 dollars from 9 apples.

48. The sign, —, is called minus, which signifies less. When placed between two numbers, it denotes that the one after it is to be taken from the one before it. Thus, 8-6 =2 is read 8 minus 6 equals 2, and shows that 6, the subtrahend, taken from 8, the minuend, equals 2, the remainder.

CASE I.

49. When no figure in the subtrahend is greater than the corresponding figure in the minuend.

1. From 574 take 323.

OPERATION.

Minuend,

574 Subtrahend, 323 Remainder, 251

ANALYSIS. Write the less number under the greater, with units under units, tens under tens, etc., and draw a line underneath. Then, beginning at the right hand, subtract separately each figure of the subtrahend from the figure above it in the minuend. Thus, 3 from 4 leaves 1, which is the difference of the units; 2 from 7 leaves 5, the difference of the tens; 3 from 5 leaves 2, the difference of the hundreds. Hence, we have for the whole difference, 2 hundreds 5 tens and 1 unit, or 251.

22203.

32343.

604400.

313241.

11. From 7634 take 3132. 12. From 41763 take 11521. 13. From 18346 take 5215.

15. Subtract 47321 from 69524. 16. Subtract 16330 from 48673. 17. Subtract 291352 from 895752. 18. Subtract 84321 from 397562.

19. A farmer paid 645 dollars for a span of horses and a carriage, and sold them for 522 dollars; what did he lose?

20. A man bought a mill for 3724 dollars, and sold it for 4856 dollars; what did he gain? Ans. 1132 dollars. 21. A drover bought 1566 sheep, and sold 435 of them; how many had he left? Ans. 1131 sheep. 22. A piece of land was sold for 2945 dollars, which was 832 dollars more than it cost; what did it cost?

23. A gentleman willed to his son 15768 dollars, and to his daughter 4537 dollars; how much more did he will to the son than to the daughter? Ans. 11231 dollars.

24. A merchant sold goods to the amount of 6742 dollars, and by so doing gained 2540 dollars; what did the goods. cost him?

25. If I borrow 15475 dollars of a person, and pay him 4050 dollars, what do I still owe him?

26. In 1850 the white population of the United States was 19,553,068, and the slave population 3,204,313; what was the difference?

27. The population of Great Britain in 1851 was 20,936,468, and of England alone, 16,921,888; what was the difference?

CASE II.

50. When any figure in the subtrahend is greater than the corresponding figure in the minuend.

1. From 846 take 359.

OPERATION.

(7) (13) (16)

Minuend,

8 4

6

5

9

4

8

Subtrahend, 3

Remainder,

ANALYSIS. In this example we cannot take 9 units from 6 units. From the 4 tens we take 1 ten, which equals 10 units, and add to the 6 units, making 16 units; 9 units from 16 units leave 7 units, which we write in the remainder in units' place. As we have taken 1 ten from the 4 tens, 3 tens only are left. We cannot take 5 tens from 3 tens; so from the 8 hundreds we take 1 hundred, which equals 10 tens, and add to the 3 tens, making 13 tens; 5 tens from 13 tens leave 8 tens, which we write in the remainder in tens' place. As we have taken 1 hundred from the 8 hundreds, 7 hundreds only are left; 3 hundreds from 7 hundreds leave 4 hundreds, which we write in the remainder in hundreds' place, and we have the whole remainder, 487.

The numbers written over the minuend are used simply to explain more clearly the method of subtracting; in practice the process should be performed mentally, and these numbers omitted.

The following method is more in accordance with practice.

OPERATION.

Hunds.

Tens.

Uuits.

846

359

ANALYSIS. Since we cannot take 9 units from 6 units, we add 10 units to 6 units, making 16 units; 9 units from 16 units leave 7 units. But as we have added 10 units, or 1 ten, to the minuend, we shall have a remainder 1 ten too large, to avoid which, we add 1 ten to the 5 tens in the subtrahend, making 6 tens. We can not take 6 tens from 4 tens; so we add 10 tens to 4, making 14 tens; 6 tens from 14 tens leave 8 tens. Now, having added 10 tens, or 1 hundred, to the minuend, we shall have a remainder 1 hundred too large, unless we add 1 hundred to the 3 hundreds in the subtrahend, making 4 hundreds ; 4 hundreds from 8 hundreds leave 4 hundreds, and we have for the total remainder, 487, the same as before.

487

Case II is what? Give explanation. Second explanation.

The process of adding 10 to the minuend is sometimes called borrowing 10, and that of adding 1 to the next figure of the subtrahend, carrying one.

51. From the preceding illustrations we have the following

RULE. I. Write the less number under the greater, placing units of the same order in the same column.

II. Begin at the right hand, and take each figure of the subtrahend from the figure above it, and write the result underneath.

III. If any figure in the subtrahend be greater than the corresponding figure above it, add 10 to that upper figure before subtracting, and then add 1 to the next left hand figure of the subtrahend.

PROOF. Add the remainder to the subtrahend, and if their sum be equal to the minuend, the work is supposed to be right.

What do we mean by borrowing 10? By carrying? Rule, first step?

Second? Third? Proof?

32. Subtract nineteen thousand nineteen from twenty thousand ten.

Ans. 991.

33. From one million nine thousand six take twenty thousand four hundred. Ans. 988606.

34. What is the difference between two million seven thousand eighteen, and one hundred five thousand seventeen?

EXAMPLES COMBINING ADDITION AND SUBTRACTION.

52. 1. A merchant gave his note for 5200 dollars. He paid at one time 2500 dollars, and at another 175 dollars; what remained due ? Ans. 2525 dollars.

2. A traveler who was 1300 miles from home, traveled homeward 235 miles in one week, in the next 275 miles, in the next 325 miles, and in the next 280 miles; how far had he still to go before he would reach home? Ans. 185 miles.

3. A man deposited in bank 8752 dollars; he drew out at one time 4234 dollars, at another 1700 dollars, at another 962