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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

16 in.; 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.

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,

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

4 b - 12 c
B, 30 20 50

66 6c 50 80 - 30

5 a 26 6c

7 a 2 a

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.

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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.

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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 ?

14 a

$1200 - $250
$500 – $350
$700 + $100

12 a – 56
5 a - 76

- 36
56

6 a

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
$600 – $250
$900 + $900

15 a + 136

5 6

13 a + 5b
4 a - 36

6 a

13 a

17 a

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

96

-86 $400 + $250

4 a + 5 6

4 a + 6 6 $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

5 6

16 a $900 – $100

4a - 2 b $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.

9a -- 6

* The answer to this problem is, $300 – $450, or - $100. Therefore, G gained $100 more than H, which means that his gains were actually 8100 less than H's.

12 a

86

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1. When one straight line meets another straight line in such a manner that two right angles are formed by the lines, the two lines are said to be perpendicular to each other.

2. Two lines side by side extending in the same direction are said to be parallel. 3. Of the lines given above: AC is

to BD BD is

to AC. MN is

to OP. OP is

to MN.

4. A line extending in the direction of the horizon is said to be horizontal. A line on the floor of the room is horizontal; a line on the ceiling is horizontal; a line on the blackboard, every point in which is equally distant from the floor, is horizontal. For convenience, lines drawn upon paper, that are parallel with the top and bottom of the paper, may be regarded as representing horizontal lines.

5. A piece of lead (plumbum), or other heavy material suspended by a cord, is called a plumb-line. A line in the direction of a plumb-line is said to be vertical. A vertical line is perpendicular to a horizontal line. Lines on the blackboard may or may not be vertical or horizontal. For convenience, lines drawn upon paper, that are parallel with the sides of the paper may be regarded as vertical lines.

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93. MISCELLANEOUS REVIEWS.

1. The angle formed by a vertical line meet- A ing a horizontal line, is an angle of de

B grees. 2. An angle that is equal to one half of a

D

С right angle, is an angle of — degrees, .

3. The angle ADB is an angle of degrees. 4. The angle BDC is an angle of de.

F grees. 5. If from a right angle, an angle of 30

G degrees be taken, the remaining angle is an angle of degrees.

6. The angle FHG is an angle of degrees.

7. During the month of November, 1897, there were consumed at the Illinois Institution for the Education of the Blind, 64 loads of coal. The weight of each load in pounds is given below. Find the total weight. 6100 8020 5490

5190
8380
6860
6800

7130
4850
6230
6560

7090
8010
6780
6690

7790
7080
6980
5780

6810
6620
6240
6980

8600
6450
6420
5990

9100
6570
6310
4740

6740
7950
6300
5520

5380
4750
6530
3630

7640
8840
6950
4930

5650
7290
6980
5150

5900
4960
4920
6420

6200
8330
5880
6770

6620
6300
7030
6220

7170
7080
5160
6020

9210

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