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68. Subtraction in arithmetic) is the process of taking one number from (out of) another.
NOTE 1.—The word number, as here used, stands for measured magnitude, or number of things.
NOTE 2.-Subtraction (in general) is the process of finding the difference of two magnitudes.
69. The minuend is the number from which another number is taken.
70. The subtrahend is the number taken from another number.
71. The difference is the number obtained by subtracting.
72. The sign –, which is read minus, indicates that the number that follows the sign is to be taken from (out of) the number that precedes it; thus, 8-3, means, that 3 is to be taken from (out of) 8.
73. PRINCIPLES. 1. Only like numbers can be subtracted.
2. The denomination of the difference is the same as that of the minuend and the subtrahend.
74. PRIMARY FACTS OF SUBTRACTION. There are eighty-one primary facts of subtraction which should be learned while learning the facts of addition. See Werner Arithmetic, Book II., p. 274, note.
7. Tell which is the minuend, which the subtrahend, and which the difference, in each of the above examples.
76. Observe that in written problems in subtraction the subtrahend is usually written under the minuend and the difference under the subtrahend; and that, as in addition, the units of the same order are written in the same column.
1. In example 2, what figures represent units of the third decimal order? of the second integral order? of the first decimal order? of the first integral order?
2. In example 5, what figures represent units of the first integral order? of the second integral order?
77. Observe that in subtraction of denominate numbers the figures that stand for units of the same denomination and order are usually written in the same column.
1. In example 4, what figures represent tens of pounds ? hundreds of pounds ?
2. In example 5, what figures represent bushels and units of the first order ?
78. Observe that in both addition and subtraction the decimal points, if there are any, usually appear in a column.
Subtraction—Simple Numbers. 79. Find the difference of 8274 and 5638. Operation.
Explanation. 8274 Eight is greater than 4. In the minuend, take one unit 5638 of the second order from the 7 units of the second order.
This unit of the second order, combined with the 4 units 2636
of the first, makes 14 units of the first order. Eight units of the first order from 14 units of the first order leave 6 units of the first order. Three units of the second order from 6 (7 — 1) units of the second order leave 3 units of the second order. Six is greater than 2.' In the minuend take one unit of the fourth order from the 8 units of the fourth order. This unit of the fourth order, combined with the 2 units of the third order, makes 12 units of the third order. Six units of the third order from 12 units of the third order leave 6 units of the third order. Five units of the fourth order from 7 (8 — 1) units of the fourth order leave 2 units of the fourth order.
The difference of 8274 and 5638 is 2636.
1. From 35642 subtract 12456.
4. From the sum of 8539, 2647, 3984, 1461, 7353, 6016, and 2364, subtract 22364.
5. From the sum of 1352, 3425, 2640, 3724, 6575, 7360, and 6276, subtract 21352.
6. From 8 thousand 1 hundred 64, subtract 3 thousand 2 hundred 75.
7. From 6 thousand 7 hundred 25, subtract 1 thousand 8 hundred 36.
8. From seven thousand four hundred sixty-five, subtract two thousand three hundred fifty-four.
(a) Find the sum of the eight differences.
TO THE PUPIL-Do not allow yourself to make one error. the eight differences and their sum, accurately, on first trial.
Subtraction-Decimals. 81. Find the difference of 28.36 and 15.432. Operation.
Explanation. 28.36 One unit of the second decimal order (1 from 6) equals 15.432 10 units of the third decimal order. Two from 10
leaves 8. 12.928
Three from 5 (6 — 1) leaves 2.
Five from 7 (8 — 1) leaves 2.
82. PROBLEMS. 1. From 100 take .3456
6. 100 – 44.764 2. From 100 take 5.246
7. 250 – 159.63 3. From 100 take 44.236 8. 250 – 36.75 4. From 100 take : 6544
9. 250 – 140.37 5. From 100 take 4.754 10. 250 – 163.25
(a) Find the sum of the ten differences.
83. MISCELLANEOUS. 1. The sum of two numbers is 3.7464; one of the numbers is 1.3521. What is the other number?
2. The difference of two nuinbers is 2.3254; the less number is 7.6746. What is the greater number?
3. The difference of two numbers is 2.3943; the greater number is 10. What is the less number?
4. The difference is 3.2678; the subtrahend is 2.1356. What is the minueud ? 5. From 10 subtract 7.6744.
(b) Find the sum of the five results.
The sum of
TO THE PUPIL.- Work with care. Make no errors. the results should be correct on first trial.
Subtraction-United States Money.
84. Find the difference of $27.25 and $14.51.
$27.25 $14.51 $12.74
85. PROBLEMS-ADDITION AND SUBTRACTION. * Note.-The following represent bank accounts of six depositors for one day. Find the sum that each depositor has to his credit at the close of the day. A.
Check $30.50 Check $12.50
Check 21.75 Check 10.80
54.20 Check 3.60
Check 18.34 Check 5.40
Check 6.24 Balance
$824.70 Check $87.50
Check $69.50 Check 89.20
Check 78.25 Check 96.40
Check 81.66 Check 94.60
(a) Find the amount of the six balances. * That bank clerk who makes one error a day in carrying out his balances, which he does not himself discover and correct, would not retain his position.