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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?

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2. C gained $1500 and $650; D gained $600 and lost How much more was C's wealth increased by the two transactions than D's?

$250.

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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?

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

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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 - $460, or -8100. Therefore, G gained $100 more than H, which means that his gains were actually $100 less than H's.

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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.

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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.

93. MISCELLANEOUS REVIEWS.

1. The angle formed by a vertical line meeting a horizontal line, is an angle of

grees.

de

B

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

right angle, is an angle of

3. The angle ADB is an angle of

grees.

D

degrees.

de

E

de

F

G

4. The angle BDC is an angle of grees.

5. If from a right angle, an angle of 30 H 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

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MULTIPLICATION.

94. Multiplication is the process of taking a number (of things) a number of times.

NOTE 1.-The word number as first used in the above statement, stands for measured magnitude. The second word number does not stand for measured magnitude, but rather for pure number, representing simply the times the number (of things) is to be repeated.

95. The multiplicand is the number (of things) taken or repeated. *

96. The multiplier is the number that shows how many times the multiplicand is to be repeated.

97. The product is the number (of things) obtained by multiplying.

98. The sign, x, which is read multiplied by, indicates that the number preceding the sign is a multiplicand, and the number following it, a multiplier.

NOTE. For other uses of this sign, see Werner Arithmetic, Book II., page 274.

99. PRINCIPLES.

1. The multiplier is always an abstract number.

2. The denomination of the product is always the same as that of the multiplicand.

100. PRIMARY FACTS OF MULTIPLICATION.

There are sixty-four primary facts of multiplication. See Werner Arithmetic, Book II., p. 275.

"The multiplicand, however written, must always be understood to express measured quantity; it is always concrete."-Psychology of Number, McClellan & Dewey, page 76.

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7. Tell which is the multiplicand, which the multiplier, and which the product, in each of the above examples.

102. Observe that in each of the above examples the multiplier is a pure number.

103. Observe that in each of the above examples the denomination of the product is the same as the denomination of the multiplicand.

104. MULTIPLICATION AND ADDITION COMPARED.

Find the sum of each of the following groups of numbers and compare the result with the product in the corresponding problem in article 101.

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