SUBTRACTION.

42. SUBTRACTION is the operation of finding the difference between two numbers.

43. The DIFFERENCE between two numbers is such a number as, added to the less, will give the greater.

44. THE MINUEND is the greater of the two numbers.

45. THE SUBTRAHEND is the less of the two numbers.

46. THE REMAINDER, or DIFFERENCE between two numbers, is the result of the operation.

47. When the two numbers are equal, either may be the minuend, and the remainder is 0.

48. Principles which control the operations.

1. The difference of two numbers added to the less number, gives the greater;

2. Like units alone can be taken from each other;

3. The difference is the same, if both numbers be equally increased.

49. Operations and Rule.

1. From 869 take 327; that is, from 8 hundreds 6 tens and 9 units, take 3 hundreds 2 tens and 7 units.

ANALYSIS.-Place the numbers so that units of the same order may fall in the same column. Beginning with the lowest order, we take units from units; then tens from tens; then, hundreds from hundreds; and find the remainder to be 542.

OPERATION.

869 min.

327 sub.

542 rem.

42. What is the difference between two numbers ?-43. What is subtraction?-44. What is the minuend?-45. What is the subtrahend?-46. What is the remainder, or difference?-47. When is the remainder 0?-48. What are the three principles that control the operations of Subtraction?-49. Give the rule for finding the difference of two numbers.

2. From the number 624 take 393.

ANALYSIS. Having written down the numbers, we subtract 3 from 4, and find a remainder 1. At the next step we meet a difficulty, for we cannot subtract 9 tens from 2 tens.

Take 1 hundred = = 10 tens, from the 6 hundreds, and add it to the 2 tens. Then, 9 tens from 12 tens, leaves 3 tens, and 3 hundreds from 5, leaves hundreds, and the remainder is 231. The remainder can be found by adding, mentally, 10 to 2 tens, and then saying, 9 from 12, leaves 3 tens; then adding 1 to 3 hundreds, and say, 4 from 6, leaves 2 hundreds.

10

624

OPERATION.

Tens.

Units.

= 5 12 4 393 = 3 9 3

1

2312

3 1

The process of adding 10 to a figure of the minuend, and returning 1 to the next figure of the subtrahend, at the left, is called borrowing.

3. From 6 T. 14 cwt. 2 qr. 20 lb. 12 oz., take 4 T. 17 cwt. 1 qr. 21 lb. 10 oz.

ANALYSIS.-Taking 10 oz. from 12 oz., 2 oz. remain. At the next step we find a difficulty, for 21 lb. cannot be taken from 20 lb. We then take 1 qr. 25 lb. from the 2 qr. and add it to the 20 lb., making 45 lb.; then say, 21 lb. from 45 lb. leaves 24 lb.; we then add 1 to the next left-hand figure of the subtrahend, and say, 2 qr. from 2 qr. leaves 0; then 17 cwt. from 34 cwt. leaves 17 cwt., and 5 from 6 leaves 1 ton.

OPERATION.

T. cwt. qr. lb. Oz.

1

1

25

20

6

14 2

20 12

I. Set down the less number under the greater, so that units of the same value shall fall in the same column :

II. Begin with the units of the lowest denomination, and subtract each number from the one above it:

III. When the number of units in any denomination of the minuend is less than in the same denomination of the subtrahend, suppose so many units to be added as make one unit of the next higher denomination; after which, add 1 to the next denomination of the subtrahend, and subtract as before.

Proof.

50. There are three methods of proving Subtraction :

I. Add the remainder to the subtrahend. right, the sum will be equal to the minuend.

If the work is

II. Subtract the remainder from the minuend. If the work is right, the remainder will be equal to the subtrahend.

III. Find the excess of 9's in the minuend, in the subtrahend, and in the remainder. If the work is right, the excess of 9's in the two last numbers will be equal to the excess of 9's in the first.

NOTE. The third method is only applicable to simple numbers.

5611 hence, the excess of 9's in the last two numbers

is 2.

Reading.

51. What is the difference between 426 and 295 ? By the common method, which is spelling, we say, 5 from 6, leaves 1; 9 from 12, leaves 3; 1 to carry to 2, are 3; 3 from 4, leaves 1.

By reading the words which express the final result, we make the operations mentally, and say, one, three, one.

OPERATION.

426 295

131

50. How many methods of proof are there? What is the first? What

the second? What the third?

51. What is spelling of numbers? What is reading?

52. How do you find the difference between two dates?

Time between Dates.

52. What time elapsed between the inauguration of Mr. Jefferson, March 4th, 12 o'clock M., 1801, and July 4th, 3 P. M., 1855?

yr.

OPERATION.

mo. da hr.

1855

7 4

15

1801

3 4 12

54

4 0 3

ANALYSIS.-Place the earlier date under the later, writing the number of the year, reckoned from the beginning of the Christian Era, on the left. Then, write in the same line the number of the month, reckoned from the first of January, the number of the day, reckoned from the first of the month, the number of the hour, reckoned from 12 at night, and write the number of minutes and seconds, if there are any, still at the right. Hence, to find the time between two dates,

Rule.-Write the earlier date under the later, and subtract as in compound numbers (Art. 49).

NOTE.-1. In finding the difference between dates, as in casting interest, the month is regarded as the twelfth part of the year, and as containing 30 days.

2. The civil day begins and ends at 12 o'clock at night.

3. If the earlier date is before the Christian Era, the sum of the numbers will express the difference of time.