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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. 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.

Five from 7 (8 – 1) leaves 2.
One from 2 leaves 1.
The difference of 28.36 and 15.432 is 12.928.

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 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. (P. 321.) 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. Operation.

Explanation.
$27.25 One cent from 5 cents = 4 cents.
$14.51

Five dimes from 12 dimes = 7 dimes.
$12.74

Four dollars from 6 dollars (7 – 1) = 2 dollars.
One ten-dollars from 2 ten-dollars =l 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. A.

B.
Deposit
$254.20
Deposit

$175.30 Deposit

38.60

Check $30.50 Check $12.50

Check 21.75 Check 10.80

Deposit

54,20 Check 3.60

Check 18.34 Check 5.40

Check 6.24
Balance

Balance
C.

D.
Deposit
$745.80 Deposit

$824.70 Check $87.50

Check $69.50 Check 89.20

Check 78.25 Check 96.40

Check 81.66 Check 94.60

Deposit

61.40 Deposit

45.80

Check 93.76
Balance

Balance
E. Deposit, $1000. Checks, $275, $324, $400.

F. Deposit, $864. Checks, $375, $146, $279. (a) Find the amount of the six balances. (P. 322.)

* 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. 10 in. Operation.

Explanation. 15 yd. 2 ft. 4 in. Ten inches are more than 4 in.; 1 ft. (from 8 yd. 1 ft. 10 in.

the 2 ft.) equals 12 in.; 12 in. and 4 in. are

10 in. from 16 in. leave 6 in. 7 yd. O ft. 6 in.

One ft. from 1 ft. (2 – 1) leaves 0 ft. Eight yd. from 15 yd. leave 7 yd. The difference of 15 yd. 2 ft. 4 in. and 8 yd. 1 ft. 10 in. is 7 yd. 6 in.

16 in.;

87. PROBLEMS.

1. From 12 yd. 1 ft. 8 in. subtract 5 yd. 2 ft. 3 in.
2. From 8 yd. 2 ft. 6 in. subtract 5 yd. 1 ft. 10 in.
3. From 9 yd. 1 ft. subtract 2 yd. 2 ft. 7 in.
4. From 6 yd. 2 ft. 5 in. subtract 4 yd. 8 in.
5. From 7 yd.

subtract 5 yd. 1 ft. 1 in.
6. From 7 yd. 1 ft. 4 in. subtract 4 yd. 9 in.
7. From 11 yd. 6 in. subtract 4 yd. 1 ft. 2 in.
8. From 10 yd. 2 ft. subtract 7 yd. 5 in.
9. From 13 yd. 7 in. subtract 5 yd. 2 ft. 4 in.
10. From 13 yd.

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. (P. 323.)

60

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.*

Mon. Tues. Wed. Thurs. Fri. Sat.
A, 80 60 20

7 a

-46 - 12c
B. 30 - 20 50 2 a - 66
50 80 - 30 5 a 26

6c 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 numbers. Find how far the first boat is from the second.

1. 2. 3. 4. 5. 6.
1st B., 75 85 15

-46 - 15 c
2nd B.,
25 - 35 35

2 a -96 50
50 120

- 20

8 a

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 num. bers losses. See page 28, Art. 54.

Algebraic Subtraction. 1. A gained $1200 and lost $250; B gained $500 and lost $350. How much more was A's wealth increased by the two transactions than B's ?

$1200 – $250 12 a-56 14 a-36
$500 - $350 5a-76

6a - 56 $ 700 + $100 2. C gained $1500 and $650; D gained $600 and lost $250. How much more was C's wealth increased by the two transactions than D's ?

$1500 + $650 15 a + 135 13 a +56
$600 – $250

6a- 56

4a - 36 $900 + $900 3. E gained $1300 and lost $450; F gained $400 and $250. How much more was E's wealth increased by the two transactions than F's ?

$1300 - $450 13 a-96 17 a-86
$400 + $ 250

4a+56 4 a +66
$900 - $ 700

4. G gained $1200 and lost $500; H gained $900 and lost $100. How much more was G's wealth increased by the two transactions than H's ? *

$1200 – $500 12 a-56 16 a-86
$900 - $100

9a - 6

4a - 26 $300 – $400

5. Review the foregoing and observe that in every instance subtracting a positive number is equivalent to adding an equal negative number, and subtracting a negative number is equivalent to adding an equal positive number.

* The answer to this problem is, $ 300 – $400, or — $ 100. Therefore, G gained — $ 100 more than H, which means that his gains were actually $ 100 less than H's.

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