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CONTRACTIONS IN ADDITION.
60. Contractions in Addition are abbreviated methods
of adding

61. There are several methods of abbreviating the pro-
cess of addition, a few of which will be stated :

1. In adding omit naming the numbers added ; merely
name the results.

2. When two or more terms of a column can be easily
grouped together, use their sum instead of adding each sep
arately; combining with especial reference to TENS.

3. When a term 18 repeated several times in a column,
multiply it by the number of times it is repeated, and use the
result.

1. What is the sum of 8752+3687+2573+8576+2857
+6872 ?

1st METHOD.—We add thus: 2, 9, 15, 18, 25, 27;
write the 7 and carry the 2; 2, 9, 14, 21, 28, 36, 41;

8752
write the 1 and carry the 4; etc.

3687
2D METHOD.-Thus: 9, 15, 25, 27, in which the 3

2573
and 7 are grouped and used as 10. We may also

group

8576
2 and 6, 7 and 3, and then 7 and 2; thus, 8, 18, 27, etc.

2857
3D METHOD.-In the second column, we may take

6872
the three 7's, then the two 5's, and then the 8; thus, 21, 33317
31, 39, etc.

OPERATION.

WRITTEN EXERCISES.
(3)
(4)

(6)
$365.75
406.26

67854
61387
57854
87543
78587
18234
34387
21857
31242
82356
14181
82815
71281
85427
18315
61038

(5)
41235 32106 $875.67
47368 21876 321.08
47376 52382 345.67
51847 36875 891.01
63529 42356 121.31
17152 52873 415.16
50381 61874 178.20
27596 18027 222.32
38273 25718 425.26
54368 43281 272.82
70305 25728 930.35
75056 71890 363.73
18293 12134 839.46
52738 51617 876.54
25382 82365 324.36
32728 57634 484.95

57.38
693.84
746.38
25.06

8.72
45.50
482.38
29.45

8.65
682.50

14.75
937.18

8.28
473.54

INTRODUCTION TO SUBTRACTION.

MENTAL EXERCISES, 1. If I have 12 books and sell 5 of them, how many books shall I bave remaining?

SOLUTION.-If I have 12 books and sell 5 of them, I have remaining 12 books minus 5 books, which are 7 books.

2. James had 14 cents and spent 9 of them; how many cents had be remaining ?

3. Susan has 25 plums and Jane 13; how many more has Susan than Jane?

4. In a school numbering 45 pupils, 15 are absent; how many pupils are present?

5. A watch was bought for 28 dollars and sold for 21 dollars; what was the loss ?

6. A cow was bought for 18 dollars and sold for 28 dollars; what was the gain?

7. Begin at 2 and count by a's to 40; begin at 40 and count by 2's backward to 2.

8. Begin at 45 and count by 3's backward to 3; begin at 44 and count by 3's backward to 2.

9. Count by 4's from 48 back to 4; from 55 back to 3; from 64 back to 2; from 53 back to 1.

10. Count by 5's from 60 back to 5; from 64 back to 4; from 63 back to 3; from 62 back to 2; from 61 back to 1.

11. In a similar manner begin at different numbers, and count backward by b's, 7's, 8's and 9's.

12. Take the number 3, add 5, subtract 6, add 7, subtract 5, add 8, subtract 7, add 9, subtract 4, and name the result.

13. Take the number 11, add 4, subtract S, add 5, subtract 4, add 6, subtract 5, add 7, subtract 6, add 8, subtract 7, and name the result

14. How many are 3 plus 5 minus 7? 4 plus 7 minus 8? 5 plus 6 minus 4? 8 plus 5 minus 9? 9 plus 10 minus 12? 12 plus 13 minus 15?

15. How many are 8 plus 12 minus 13? 9 plus 16 minus 14? 10 plus 15 minus 16? 16 plus 17 minus 18? 18 plus 19 minus 20?

The process of finding the difference between two numbers is called subtraction. The sign of subtraction is -, and is read minus.

16. Required the value of the following: 5+842 275-4

978-6 | 107 7—12 13+14-15 6+4-3 B76-5 876—7 11+ 8-13 14+15–17 7+244 447-8 7+5–4 12+ 7-9 15+18-19 87546 B+847

6+8–8 8+14-10 18+17-12 918-5 8+7-9 5+9–4 9+16–16 17+14--20

SUBTRACTION.

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

63. The Difference between two numbers is a number which added to the less, equals the greater.

64. The Minuend is the number from which we subtract he Subtrahend is the number to be subtracted.

65. The Sign of Subtraction is and is read minus It denotes that the number immediately following it is to be subtracted from the number preceding it.

NOTES 1. The Sign of Subtraction is a short line in the line of writing.

2. The symbol was introduced by Stifelius, a German mathematician, In a work published in 1544.

PRINCIPLES, 1. The minuend and subtrahend must be similar numbers.

2. The difference is a number similar to the minuend and subtrahend.

CASE I. 66. To subtract when no term of the subtrahend is greater than the corresponding term of the minuend.

1. What is the difference between 486 and 243 ? SOLUTION.- We write the subtrahend under the minuend, placing terms of the same order in the same col- 486 umn, draw a line beneath, and begin at the right to 243 subtract. 3 units from 6 units leave 3 units, which we

243 write under the units ; 4 tens from 8 tens leave 4 tens, which we write under the tens; 2 hundreds from 4 hundreds leave 2 hundreds, which we write under the hundreds Therefore, the difference between 486 and 243 is 243.

Rule.-I. Write the subtrahend under the minuend, placing terms of the same order in the same column, and draw a line beneath.

II. Begin at the right and subtract each term of the sub trahend from the corresponding term of the minuend, vomit ing the remainder beneath.

OPERATION.

WRITTEN EXERCISES.

From
Subtract

364
123

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67. To subtract when one or more terms of the subtrahend are greater than the corresponding terms of the minuend.

68. There are two methods of explaining this case, called the Method of Borrowing and the Method of Adding Ten. We will solve the same problem by both methods

1. From 836 subtract 472. SOLUTION BY BORROWING.–We write the subtrabend under the minuend, and begin at the right to sub- 836 tract. 2 units from 6 units leave 4 units, which we 472 write under the units; we cannot take 7 tens from 3

364 tens, we will therefore take 1 hundred from the 8 hundreds, and add it to the 3 tens; 1 hundred equals 10 tens, which, added to 3 tens, equals 13 tens; 7 tens from 13 tens le&ve 6 tens, which we write in tens place ; 4 hundreds from 7 handreds (the bumher remaining after taking away 1 hundred) leave 3 hundrede, which we write in the hundreds place.

OPERATION.

SOLUTION BY ADDING TEN.—2 units from 6 units leave 4 units; wo annot take 7 tens from 3 tens, we will therefore add 10 tens to the 3 tens, making 13 tens; 7 tens from 13 tens leave 6 tens; now, since we have added 10 tens, or 1 hundred, to the minuend, our remainder will be 1 hundred too large; hence to obtain the correct remainder we must add 1 hundred to the subtrahend; 1 hundred and 4 hundreds are 8 hundreds; 5 hundreds from 8 hundreds leave 3 hundreds.

Rule.-I. Write the subtrahend under the minuend, plac ing terms of the same order in the same column, and draw a line beneath.

II. Begin at the right and subtract each term of the sube trahend from the corresponding term of the minuend, writing the remainder beneath.

III. If any term of the subtrahend is greater than the corresponding term of the minuend, add 10 to the latter, and then subtract.

IV. Add 1 to the next term of the subtrahend (or subtract 1 from the next term of the minuend), and proceed as before.

Proof.-Add the difference to the subtrahend; and if the work is correct the sum will equal the minuend.

SECOND METHOD.-Subtract the difference from the minuend, and if the work is correct, the result will equal the subtrahend.

NOTE.—The taking one from a term of the minuend is called borrowing, and the adding one to the next term of the subtrahend is called carrying.

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