SUBTRACTION.

53. If you have five marbles in your hand and take away two of them, how many marbles have you left?

Taking two marbles, a part of five marbles, away, we find there are three marbles left.

The process of taking part of a number away to find how many are left is Subtraction. What is subtraction?

54. The number, a part of which is taken away, is the minuend.

Which is the minuend in the given illustration?

55. The part of the minuend taken away is the subtrahend.

What is the subtrahend?

Which is the subtrahend in the given illustration?

56. That part of the minuend left, after a part is taken away, is the remainder.

What is the remainder?

What is the remainder in the given illustration?

57. ORAL EXERCISES.1. If you have 11 apples, and give 8 of them away, how many remain? How many remain if you give away 2? 4? 9? 5? 3? 6? 7?

2. If there are 12 pears and you take away 5 of them, how many pears remain? How many remain if you take away 9? 7? 6? 8? 4? 3?

3. Mary has 13 peaches, and Henry has 4 less than Mary; how many has Henry? How many are 13 less 4? 13 less 5? 13 less 8? 13 less 7? 13 less 6? 13 less 9?

4. How many are 14 less 8? 14 less 5? 14 less 7? 14 less 6?

15; if one

14 less 9?

5. The sum of two numbers is of the numbers is 7, what is the other? If one is 6, what is the other? If one is 9, what is the other? If one is 8, what is the other?

6. 8 oranges and how many are 16 oranges? 7 and how many are 16? 9 and how many are 16?

7. 17 are how many more than 8? than 9? how many more than 9?

18 are

58. ILLUSTRATIVE EXAMPLE I. If Henry has 5 apples and Mary has 3, how many more apples has Henry than Mary?

We may find how many more Henry has by taking away a number of his apples equal to Mary's. 3 of Henry's apples taken leaves 2.

We see, then, that Henry has 2 apples more than Mary has, or that the difference of the numbers 5 and 3 is 2.

Hence to find the difference of two numbers, we take away a part of the larger number equal to the smaller.

This process is the same as that illustrated in Art. 53, and is therefore Subtraction.

How many more are 17 eggs than 9 eggs? Tell how you know this?

59. The process of subtraction may be thus expressed: 6 marbles, whole number, called minuend. 2 marbles, one part given, called subtrahend.

4 marbles, other part found, called remainder.

REMARK. — On comparing subtraction with addition, it will be seen that one is the reverse of the other, for while in addition we increase a number by the number added, in subtraction we diminish a number by the number taken away.

60. The sign of subtraction is a short horizontal mark made thus, and read "minus' or "less."

pression

The ex

shows that if 2 of 5 cents are taken away, 3 cents will remain.

Read and explain the expressions

61. ILLUSTRATIVE EXAMPLE II. Jane had 27 apples, and gave 13 of them to Mary; how many apples had she left?

Tens.

Units.

OPERATION.

27 whole number.
13 part given.
14 part remaining.

Explanation. To find how many

she had left, we take 13 of the number 27 away. We express the number 27 and 13 under it so that units of the same order shall be expressed in the same column, and draw a line underneath.

27 equals 2 tens with 7 units; 13 equals 1 ten with 3 units.

If 3 of the 7 units are taken away, 4 units remain; * we write a figure 4 under the line in the units' place.

If 1 of the 2 tens is taken away, 1 ten remains; we write a figure

*TO THE TEACHER. - This expression is preferable to the expression, "If 3 units are taken from 7 units," etc., since "from" commonly denotes "leaving behind" or "departure" rather than "out of."

If 3 men should walk away from 7 men, 7 men would be left, and not 4 men; but if 3 of 7 men should walk away from the rest, 4 would be left. "If three units are taken out of 7 units" is a correct and convenient form of expression.

1 in the tens' place, and have 1 ten with 4 units, or 14, for the entire remainder, which is the number of apples Jane had left.

NOTE.. The above illustrates the manner of explaining an example in subtraction; but in practice the pupil needs only say "3 of 7 taken leaves 4; 1 of 2 taken leaves 1;" or "3 out of 7 leaves 4; 1 out of 2 leaves 1."

62. PROOF OF ILLUSTRATIVE EXAMPLE II.

13 part given.
14 part remaining.
27 whole number.

The number of apples Jane has left, or 14, added to the number she gave away, or 13, must equal the number she had at first; adding them, we have 27 for the result; we may therefore conclude that the work was correct; and, generally,

To prove the work of any example in subtraction, add the remainder to the subtrahend; if the work is correct, their sum must equal the minuend.

63. EXAMPLES.

8. John had 28 cents, and spent 17 of them for a slate; how many cents had he left? Ans. 11 cents.

9. Albert had 56 cents, and spent 32 of them for a hat, and the rest for a book; how many cents did he spend for the book? Ans. 24 cents.

10. Olive earned 98 cents in a day, and Sarah earned 72 cents; how many more cents did Olive earn than Sarah?

NOTE. - She earned as many more than Sarah as 98 cents are more than 72 cents, which may be found by taking 72 of the number 98 away. Ans. 26 cents.

11. Mr. Goss paid 275 dollars for his horse, and 124 dollars for his oxen; how many more dollars did he pay for his horse than for his oxen? Ans. 151 dollars.

12. How many more sheep are there in a flock of 389 sheep than in a flock of 278 sheep? Ans. 111 sheep.

13. If of 278 bricks 235 bricks are taken away, how many bricks remain? Ans. 43 bricks.

14. If of a regiment of 785 men 572 were taken prisonhow many remained? Ans. 213 men.

ers,

15. If of a drove of 1648 cattle 524 are sold, how many will remain ?

NOTE. When the answer is not given, the pupil should prove his work. (Art. 62.)

16. In the schools of a certain town, in one year there were 3697 cases of tardiness; the next year there were 2253 less; how many were there in the second year?

64. ILLUSTRATIVE EXAMPLE III. Annie had 243 plums, and gave Arthur 125 of them; how many plums had she left?

OPERATION.

Hundreds.

(3)(13)

243
125
118

Explanation.

As there are but 3 units in the minuend, we cannot take 5 units away; we will therefore change one of the 4 tens (leaving 3 tens) to units. 1 ten equals 10 units; ten units with 3 units are 13 units. If 5 of these 13 units are taken away, 8 units remain.

If 2 of the 3 tens are taken away, 1 ten remains. If 1 of the 2 hundreds is taken away, 1 hundred remains. We have, therefore, 118 for the entire remainder; which is the number of plums Annie had left.

NOTE..

In practice, the pupil needs only say 5 of 13 taken leaves 8; 2 of 3 taken leaves 1," etc, or "5 out of 13 leaves 8, 2 out of 3 leaves 1,"