23. Adding Two or More Columns at Once.

When small numbers are to be added, it is sometimes convenient to add two or even three columns at one time.

In the example in the margin, we say: 46, 66, 70, 160, 161, 201, 207. That is, we add 46 and 20, getting 66; then 66 and 4, getting 70; then 70 and 90, getting 160, then 160 and 1, getting 161; then 161 and 40, getting 201; then 201 and 6, getting 207.

ORAL EXERCISES

In this manner add each of the following:

46

24

91

46

207

24. Horizontal Addition of Small Numbers. The method used in adding two or more columns at a time is useful in horizontal addition, since it avoids the danger of adding numbers of different order. That is, 28 +45 +61 + 35 + 78 + 16 We say, 28, 68, 73, 133, 134, 164, 169, 239, 247, 257, 263.

Or, 28, 73, 134, 169, 247, 263.

= 263.

ORAL EXERCISES

In this manner add the following: 1. 73 + 21 + 67 + 82 + 35 + 61. 2. 14 + 18 + 16 + 23 + 7 + 27. 3. 15 + 17 + 12 + 16 + 14 + 19. 4. 24 + 18 + 9 + 17 + 16 + 15. 5. 25 +37 + 91 + 8 + 24 + 15. 6. 14+7+ 21 + 42 +63 +91.

7. 22 +44 + 88 + 17 + 18 + 26.

8. 24+ 96 + 8 + 127 + 225.

9. 64+ 125 +216 + 343 + 512 + 1728.

EXERCISES

Add the following by grouping as on page 9. Write down the results only.

Add the following as on page 17. Write down the results only. Do not copy the example.

Add the following as in § 23. Do not copy the examples. Write down results only.

10. 41 + 64 + 31 + 14 + 71 + 24 + 62. 11. 1920 +46 +39 + 78+22+61. 12. 26+35+ 19 +17 + 66 + 42 + 18. 13. 47 +39 + 62 + 45 + 20 + 76 + 33. 14. 1876 + 23 + 48 + 18 + 81 + 46. 15. 42 +91 +27 + 15 +56 + 73 + 38. 16. 6587 + 23 + 39 + 22 + 38 + 17. 17. 91 + 15 + 26 + 14 + 17 + 83 + 67. 18. 76 +23 + 10 + 23 +93 + 77 + 29. 19. 28 +55 + 26 +95 +49 + 26 + 35. 20. 17 +21 +36 + 68 + 38 + 19 + 13.

CHAPTER III

SUBTRACTION

25. Definition of Subtraction. Subtraction is the process of finding how much must be added to a given number to make the sum equal to another given number.

Thus: "How much must be added to 30 to make the sum 70?" is a problem in subtraction.

26. Minuend, Subtrahend, Remainder. The given sum is called the minuend and the other given number is called the subtrahend. The number to be found is called the remainder.

The sign

indicates subtraction and is read minus or less. The number before the sign is the minuend, and the number after it is the subtrahend. The expression" 70 - 30 = 40," is read "70 minus 30 equals 40."

We also say that “Sub

27. Other Definitions of Subtraction. traction is the process of finding the difference between two numbers," or that "it consists in finding how much is left when one number is taken from another number.”

All these definitions of subtraction amount to the same thing. Thus we may ask, "What is the difference between 30 and 70? or "How much must be added to 30 to make 70?" or "How much is left when 30 is taken from 70?"

In this book subtraction will be regarded as the process of finding how much must be added to one number, the subtrahend, to give another number, the minuend. The process of subtraction will be performed by actually adding enough to the subtrahend to produce the minuend.

This method of subtraction has the advantage that it connects more directly with addition than the other methods.

66

28. Uses of Subtraction. - Subtraction is used constantly in business. Making change " is one of its simplest uses. Finding balances of accounts, gains or losses, etc., all involve subtraction.

29. Subtraction without Carrying. tract 3235. 8647

Example. From 8647 sub

Write the numbers so that figures of the same order are in the 3235 same column. The subtrahend is usually written below the minu5412 end. Then find what number must be added to the subtrahend to produce the minuend. This number is written below the line. We say, 7 = 5 + 2, 4 2 + 4, 8 = 3+ 5.

=

3+1, 6

=

With sufficient practice one sees the differences by a mere glance at the figures. These can be written as rapidly as one can conveniently write figures.

30. Subtraction with Carrying.

Example 1. From 3924 subtract 2718.

=

Since 4 is not the sum of 8 and any other number, we add 10 to 4 and say 14 86. Write 6, carry 1 (ten) to the subtrahend. (Adding 10 to the minuend and carrying 1 (ten) in the subtrahend adds the same number to both subtrahend and minuend and hence leaves the remainder unchanged.)

1 (carried) + 1 + 0.

Write 0.

2+1, completes the subtraction.

Example 2. From 34,001 subtract 17,268.