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57. A circle is a plane figure bounded by a curved line, every point in which is equally distant from a point within, called the center.
58. The line that bounds a circle is a circumference.
59. A straight line passing through the center of a circle and ending in the circumference is a diameter.
60. A straight line from the center of a circle to the circumference is a radius.
61. Any part of a circumference is an arc.
62. For the purpose of measurement, every circumference is considered as divided into 360 equal parts, called degrees.
g 63. Two lines meeting at a point form an angle. The point in which the two lines meet is the vertex of the angle.
64. Every angle may be regarded as having its vertex at the center of a circle, and the angle is measured by the 1
k part of the arc intercepted; thus, the angle hlk is measured by the arc mn.
65. The angle efg is an angle of 90 degrees, called also a right angle. The angle abc is an angle of 90 degrees. The angle hlk is an angle of 45 degrees.
66. All the angles about a point together equal four right angles.
67. MISCELLANEOUS REVIEW. 1. The angle abd is an angle of about -degrees. The angle dbc is an angle of — degrees.
d The angle abd + the angle dbc = the angle abc. The angle abc is an angle of -- degrees.
b 2. The angle mln is an angle of about – degrees.
The angle nls is an angle of about - degrees.
The angle min + the angle nls = the angle mls.
1 The angle mls is an angle of about — degrees. 3. Earnings of ten persons for six days.
Copy the following figures and add by column and by line. Prove by comparing the sum of the sums of the columns with the sum of the sums of the lines. That pupil who can solve this problem without an error, on first trial, has taken an important step toward making himself valuable as an accountant.
Mon. Tues. Wed. Thurs. Frid. Sat.
5.05 5.15 4.95
2.06 2.25 3.50 3.60 3.40 3.30 3.50 3.60 3.05 2.90 3.15 2.95 3.15 2.00
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.
75. EXAMPLES OF SUBTRACTION.
1. 2687 1298 1389
2. 57.38 28.146 29.234
25 a +66
10 a +26
15 a +46 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. 2636
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.
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. Find the eight differences and their sum, accurately, on first trial.