40. The railroad distance from New York to Albany is 144 miles; from Albany to Buffalo, 298; from Buffalo to Cleveland, 183; from Cleveland to Toledo, 109; from Toledo to Springfield, 365; and from Springfield to St. Louis, 95 miles; what is the distance from New York to St. Louis? 41. A man owns farms valued at 56800 dollars; city lots valued at 86760 dollars; a house worth 12500 dollars, and other property to the amount of 6785 dollars; what is the entire value of his property? Ans. 162845 dollars.

SUBTRACTION:

MENTAL EXERCISES.

43. 1. A farmer, having 14 cows, sold 6 of them; how many had he left ?

ANALYSIS. He had as many left as 14 cows less. 6 cows, which are 8 cows.

2. Stephen, having 9 marbles, lost 4 of them; how many had he left?

3. If a man earn 10 dollars a week, and spend 6 dollars for provisions, how many dollars has he left?

4. A merchant, having 16 barrels of flour, sells 9 of them; how many barrels has he left?

5. Charles had 18 cents, and gave 10 of them for a book; how many cents had he left ?

6. James is 17 years old, and his sister Julia is 5 years younger; how old is Julia ?

7. A grocer, having 20 boxes of lemons, sold 11 boxes; how many boxes had he left?

8. From a cistern containing 25 barrels of water, 15 barrels leaked out; how many barrels remained?

9. Paid 16 dollars for a coat, and 7 dollars for a vest; how much more did the coat cost than the vest?

10. How many are 18 less 5? 11. How many are 20 less 14? 12. How many are 11 less 3? 20 less 9? 22 less 20?

17 less 8? 12 less 7?

18 less 12?

19 less 11?

16 less 11?

19 less 8?

44. Subtraction is the process of finding the difference between two numbers of the same kind.

45. The Minuend is the number to be subtracted from. 46. The Subtrahend is the number to be subtracted.

Define subtraction? Minuend? Subtrahend?

4. 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 dollars 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.

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, 16,921,868; 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

9

4

8

7

Subtrahend, 3 5

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.

T'ens.

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.