EXPLANATION.-Since it is customary to read from left to right, we begin at the left with the units figure, 4, of the first number, and add to it successively the units 6, 4, 3, and 1 of the other numbers, making 18. We write down the 8 and carry the 1 to the ten's figure, 6, of the first number, and add to it successively the tens 5, 5, 2, and 3 of the other numbers, making 22. We write down the 2 and carry the other 2 to the hundred's figure, 3, of the first number, and add to it successively the hundreds 1, 7, and 6 of the other numbers containing hundreds, making 19. We write down the 9 and carry the 1 to the thousand's figure 2, of the first number, and add it to the thousand's figure, 1, of the fourth number (the other numbers not having a thousand's place), making 4, which we write. Hence, the sum of the numbers is 4928.

Copy and add horizontally the following:

2. 4678, 537, 642, 40354, 6342, 96.
3. 734, 865, 732, 672, 8904, 6742.

4. 1043, 27, 462, 8409, 76431, 27, 4896.
5. 720, 42, 76, 871, 5630, 4725, 87.
6. 1726, 307, 4268, 9725, 32, 403, 26.
7. 34268, 28, 135, 6724. 5962, 306.
8. 59260, 172, 427, 3964, 7685, 703.
9. 14032, 675, 1089, 7638, 976472.

10. 89642, 4235, 72168, 73, 5, 987.

11. 64356, 742345, 67345, 9643, 756, 7856, 723.
12. 72, 814, 1734, 6896, 4672, 596, 754, 9147.
13. 7627, 42306, 76248, 964, 505, 273, 45, 1005.
14. 53678, 904, 672, 96, 7, 49, 365, 27, 652.
15. 345, 768, 9764, 642, 567, 8952, 354, 1024.

16. Add vertically and horizontally the following statement of receipts:

Total...

Mon. Tues. Wed. Thurs. Friday. Sat. Total.

$732.14 $462.48 $800.45 $100.36 $1000.50 $465.30

10.45 103.24 74.34 9.46

16.76 16.35 85.16 972.18

741.16 786.52

806.30 394.15

SUBTRACTION.

62. Subtraction is the process of finding the difference between two numbers.

63. The Minuend is the number from which we subtract.

64. The Subtrahend is the number to be subtracted, or taken from the minuend.

65. The Difference, or Remainder, is the result obtained after subtracting.

66. The Sign of Subtraction is a short horizontal line, and is read minus. It denotes that the number after it is to be taken from the number before it.

3 is read 7 minus 4 equals 3, and means that 4 subtracted

67. PRINCIPLES.-1. Only like numbers can be subtracted.

2. The difference or remainder must be like the minuend and subtrahend.

3. The sum of the remainder and the subtrahend is equal to the minuend.

EXPLANATION.-This table is made the same as the addition table. The numbers in the left-hand column may be regarded as the subtrahend, and the numbers in the horizontal columns opposite the subtrahends as the minuends. The remainders will be found across the top.

Thus, 1 from 1 leaves 0; 1 from 2 leaves 1; 1 from 3 leaves 2; 1 from 4 leaves 3; 1 from 5 leaves 4, etc.

The order may be changed so that the numbers across the top may be regarded as the subtrahends, and the numbers on the left as the remainders. The pupil should work out the remainders for himself, and then thoroughly master the table.

DRILL EXERCISES.

68. The student should practice these exercises daily until he can tell the results instantly.

WRITTEN EXERCISES.

69. 1. Subtract 245 from 698.

Minuend, 698
Subtrahend, 245
Remainder, 453

EXPLANATION.-For convenience, we write the subtrahend under the minuend, placing figures of the same order in the same column. We draw a line beneath and begin with units to subtract. 5 units from 8 units leave 3 units, which we write under the units; 4 tens from 9 tens leave 5 tens, which we write under the tens; 2 hundreds from 6 hundreds leave 4 hundreds, which we write under the hundreds. Hence, the difference, or remainder is 453.

PROOF.-453, the remainder, plus 245, the subtrahend, equals 698, the minuend. Hence, the result in correct (Prin. 3).

2. From 852 subtract 437.

852

METHOD BY BORROWING.-We write the subtrahend under the 437 minuend as before, and begin at the right to subtract. We cannot 415 take 7 units from 2 units; but we can take 1 ten, which is equal to 10

units, from the 5 tens and unite it with the 2 units, making 12 units.

7 units from 12 units leave 5 units, which we write under the units.

Since 1 ten was taken from the 5 tens, there are but 4 tens left. 3 tens from 4 tens leave 1 ten, which is written under the tens. 4 hundreds from 8 hundreds leave 4 hundreds, which are written under the hundreds. Hence, the remainder is 415.

PROOF.-415, the remainder, plus 437, the subtrahend, equals 852, the minuend. Hence, the result is correct.

METHOD BY ADDING TEN.-Since 7 units cannot be subtracted from 2 units, we will therefore add 10 units to the 2 units, making 12 units. 7 units from 12 units leave 5 units. Now, since we have added 10 units, or 1 ten, to the minuend, our remainder will be 1 ten too large. Hence, to secure the correct remainder, we must add 1 ten to the subtrahend.. 1 ten and 3 tens are 4 tens. 4 tens from 5 tens leave 1 ten; and 4 hundreds from 8 hundreds leave 4 hundreds. Hence, the remainder is 415.

In practice, we say, 7 from 2, impossible, but 7 from 12 leaves 5, 3 from 4 (or 4 from 5) leaves 1, and 4 from 8 leaves 4. Answer, 415.

RULE. Write the subtrahend under the minuend, so that figures of the same order stand in the same column, and draw a line beneath.

Begin at the right, subtract the number expressed by each figure of the subtrahend from the number expressed by the figure above it, and write the difference beneath.

If any figure of the subtrahend express a greater number than the figure above it, add ten to the latter, and then subtract. Take one from the number expressed by the next figure of the minuend or

add one to the number expressed by the next figure of the subtrahend, and proceed as before.

PROOF. Add the remainder to the subtrahend and, if the work is correct, the sum will equal the minuend.