Subtraction-Simple Numbers.

79. Find the difference of 8274 and 5638.

Operation. 8274 5638 2636

Explanation.

Eight is greater than 4. In the minuend, take one unit of the second order from the 7 units of the second order. This unit of the second order, combined with the 4 units 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.

80. PROBLEMS.

1. From 35642 subtract 12456.
2. From 87544 subtract 64358.

3. From 90070 subtract 13256.

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. Find the eight differences and their sum, accurately, on first trial.

Subtraction-Decimals.

81. Find the difference of 28.36 and 15.432.

Operation. 28.36

Explanation.

One unit of the second decimal order (1 from 6) equals 15.432 10 units of the third decimal order. Two from 10

Four is greater than 3. One unit of the first integral order (1 from 8) equals 10 units of the first decimal order. 10+ 3 = 13. Four from 13 leaves 9.

(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 numbers 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 minuend?

5. From 10 subtract 7.6744.

(b) Find the sum of the five results.

TO THE PUPIL.-Work with care. Make no errors. The sum of the results should be correct on first trial.

Subtraction-United States Money.

84. Find the difference of $27.25 and $14.51.

$12.74

Four dollars from 6 dollars (7 −1) = 2 dollars.
One ten-dollars from 2 ten-dollars: = 1 ten-dollars.
The difference of $27.25 and $14.51 is $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.

E. Deposit, $1000.

F. Deposit, $864.

Checks, $275, $324, $400.

Checks, $375, $146, $279.

(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.

Subtraction-Denominate Numbers.

86. Find the difference of 15 yd. 2 ft. 4 in. and 8 yd. 1 ft.

The difference of 15 yd. 2 ft. 4 in. and 8 yd. 1 ft. 10 in. is 7 yd. 6 in.

87. PROBLEMS.

1. From 12 yd. 1 ft. 8 in. 2. From 8 yd. 2 ft. 6 in. 3. From 9 yd. 1 ft.

subtract 5 yd. 2 ft. 3 in.
subtract 5 yd. 1 ft. 10 in.
subtract 2 yd. 2 ft. 7 in.

4. From 6 yd. 2 ft. 5 in. subtract 4 yd. 8 in.
5. From 7 yd.

6. From 7 yd. 1 ft. 4 in.
7. From 11 yd. 6 in.
8. From 10 yd. 2 ft.

9. From 13 yd. 7 in.

10. From 13 yd.

subtract 5 yd. 1 ft. 1 in.
subtract 4 yd. 9 in.
subtract 4 yd. 1 ft. 2 in.
subtract 7 yd. 5 in.
subtract 5 yd. 2 ft. 4 in.
subtract 4 yd. 1 ft. 11 in.

(a) Find the sum of the ten differences.

88. PROBLEMS.

1. How many days from April 25 to May 1?
2 How many days from April 25 to May 10?
3. How many days from April 25 to May 20?
4. How many days from April 25 to June 1?
5. How many days from April 25 to June 10?
6. How many days from April 25 to June 30?
7. How many days from April 25 to July 5?
8. How many days from April 25 to Sept. 10?
(b) Find the sum of the eight answers.

Algebraic Subtraction.

89. Regarding the following minuends as representing A's gain (or loss), and the subtrahends as representing B's gain (or loss), subtract B's from A's.*

NOTE.-The positive differences for Monday, Tuesday, Thursday, and Friday indicate that A's gain was greater (or his loss less) than B's. The negative differences for Wednesday and Saturday indicate that A's gain was less (or his loss greater) than B's.

90. Regard the following minuends as representing distances one boat sails from a given point, and the subtrahends as representing distances another boat sails from the same point. Distances sailed north are here represented by positive numbers, and distances sailed south by negative num. bers. Find how far the first boat is from the second.

NOTE.-The positive differences in Nos. 1, 2, 4, and 5 indicate that the first boat is north of the second boat. The negative differences in Nos. 3 and 6 indicate that the first boat is south of the second boat.

91. From the foregoing learn that the subtraction of a positive number gives the same result as the addition of an equal negative number; and the subtraction of a negative number the same result as the addition of an equal positive number. Hence the rule for algebraic subtraction: Conceive the sign (or signs) of the subtrahend to be changed (— to + and + to -), then proceed as in addition.

* Remember that positive numbers are to represent gains and negative numbers losses. See page 28, Art. 54.