Arithmetic for Schools

SUBTRACTION.

33. The process of finding how many units are left when a number is taken away from a larger number is called Subtraction. The result is called the Remainder,

or the Difference.

Any two numbers can be added; it is, however, impossible to subtract one number from a smaller number.

34. The larger of the two numbers is called the Minuend.

The smaller of the two numbers is called the Subtrahend.

35. It is clear that the remainder is that number which, when added to the subtrahend, will give the minuend.

Thus, to subtract 5 from 12 is to find the number which, when added to 5, will make 12.

The question involved in subtraction may be put in different ways. Thus, it may be asked:

(1) What is the remainder when 5 is taken from 12? (2) What must be added to 5 to make 12?

(3) By how many is 12 greater than 5?

(4) By how many is 5 less than 12?

36. Subtraction is indicated by the sign which is read minus.'

Thus, 9 4 is read nine minus four, and denotes that 9 is to be diminished by 4, that is, that 4 is to be subtracted from 9; also, 54+ 3 denotes that 4 is to be taken from 5, and then 3 added to the result.

37. The knowledge of the results of the addition of numbers not greater than ten will furnish us with the

166 respectmely !

2. HV

ET In kâ vien ve ake 5 e 14.

4 from 13, 8 fom 167 a 1)fm 11. and 5 frum 13, respectively?

Find the difference letreen the numbers in each of the following pairs:

3. 5 and 12. 7 and 14 9 and 18. 3 and 11.6 and 14. 8 and 15.

4. 3 and 8.5 and 11. 6 ani 13. S and 14.7 and 15, 9 and 16.

5. Begin with 50 and go en diminishing by fours as many times as possible.

6. Begin with 53 and go on diminishing by fives as many times as possible.

7. Begin with 70 and go on diminishing by sixes as many times as possible.

8. What must be added to 5 to make 8, to make 13, to make 10, to make 12?

9. What must be added to 7 to make 9, to make 12, to make 10, to make 15?

10. What must be added to 8 to make 10, to make 12, to make 14, to make 16?

38. The consideration of the following examples will show how the difference between any two numbers can be found.

Ex. 1. Subtract 524.63 from 759.85.

The smaller number should be placed just under the greater, so that one decimal point is vertically over the other. (See Art. 29.)

759.85
524.63

Beginning with the lowest order, we find the remainder when 3 hundredths are taken from 5 hundredths, 6 tenths from 8 tenths, 4 units from 9 units, 2 tens from 5 tens, and 5 hundreds from 7 hundreds; thus,

Now 7 tenths are more than 3 tenths, therefore we cannot subtract: if, however, we take 1 unit from the 8 units and change that unit to 10 tenths, we shall have 13 tenths in all. Now 7 tenths from 13 tenths leave 6 tenths, 5 units from 7 units leave 2 units, and 3 tens from 7 tens leave 4 tens. Remainder = 42.6.

Mental Work Illustrated. We may omit names of orders. (See note, Art. 29.

In this example 1 ten is taken from 2 tens and changed to 10 units; one of these units is changed to ten tenths. The operation may be represented thus:

39. One concrete number cannot be subtracted from another unless both are expressed in terms of the same unit. For example, we cannot subtract 5 tons from 7 miles; nor can we subtract 3 feet from 60 inches, unless either 3 feet is expressed in inches or 60 inches expressed in feet.

40. It is easily seen that if from a given number several numbers be taken in succession the result will be the same as if the sum of those numbers were subtracted from the given number.

Ex. Subtract the sum of 366, 648, and 759 from 2314.

2314

366

648 759 541

9, 8, and 6 make 23; subtract the 3 from the 4 and carry the 2; 2, 5, 4, and 6 make 17; subtract the 7 from 11 and carry the 1; 1, 7, 6, and 3 make 17, which is to be subtracted from 22.

41. When several operations of addition and subtraction have to be performed in succession the result is the same in whatever order the operations are performed.

Hence, to find 28 15+ 26 - 17

1412, first find the sum of

28, 26, and 12, the numbers to be added; then the sum of 15, 17, and 14, the numbers to be subtracted; and finally taking the difference of these two sums; thus,

42. To detect mistakes in subtraction, add the remainder to the subtrahend, and the sum should equal the minuend; or subtract the remainder from the minuend, and the new remainder should equal the subtrahend.

EXAMPLES VII.

Written Exercises.

1. Subtract 129.6 from 3145, 81.7 from 3002, and 123.4 from 432.1.

2. Subtract 15.97 from 79.15, 18235 from 1000000, and 135.79 from 24680.6.

3. Find the values of 645 – 378, 307 - 149, 294 208, 2179 1984, 3206-1679, and 120573 - 98765.

Find the difference between

4. 3.726 and 5.949.

5. 14.753 and 6.876.

6. 1 and .888.

7. .00013 and .00175.

12. Find the values of