Explanation. The sum of the number of quarts is 20; this is equal to 2 pecks and 4 quarts. Write the 4 quarts and add the 2 pecks to the pecks given in the second column. The sum of the number of pecks is 11; this is equal to 2 bushels and 3 pecks. Write the 3 pecks, and add the 2 bushels to the bushels given in the third column. The sum of the number of bushels is 27, which is written in its place. The sum is 27 bu. 3 pk. 4 qt. Algebraic Addition. 47. A coefficient is a number that indicates how many times a literal quantity* is to be taken; thus, in the expression 4a6, 4 is the coefficient of a&.f When no coefficient is expressed, it is understood that 1 is the coefficient; thus, in the expression 4a+ 6, the coefficient of 6 is 1. 48. The terms of an algebraic expression are the parts that are separated by the sign + or —. There are three terms in the following: ab -+- 3c + iabc. There are only two terms in the following: 8ax45+5a + 66. 49. Positive terms are usually preceded by the plus sign. 50. Negative terms are preceded by the minus sign. If no sign is expressed, the term is understood to be positive. 51. When the literal part of two or more terms is the same, the terms are said to be similar. 52. Problems. Unite the terms in each of the following algebraic expressions into one equivalent term: 1. 5a;+3a; + 2a; = 5. iab + lab + Sab = 2. 4x + 5x - 3x = 6. lab + 5a6 - 4ab = 3. 6a - 2a + 4a = 7. 36c - 5bc + 66e = 53. Language Exercise. Referring to the problems given above, use the following words in complete sentences: Coefficient, terms, positive, negative, similar, literal. *The word quantity in algebra means number. The expression literal quantity means number expressed by letters. t The term coerBcient is sometimes applied to the literal part of an expression: thus, in the expression abc, ab is the coefficient of c. Usually, however, the term coefficient has reference to the numerical coefficient. Algebraic Addition. 54. Regarding the following positive numbers as representing gains and the negative numbers as representing losses, find the total gain (or loss) in each case; that is, find the algebraic sum of the numbers in each group: Note.—The positive sums of Nos. 1, 2, 4, and 5 indicate actual gain; the negative sums of Nos. 3 and 6 indicate actual loss. 55. Regarding each of the following positive numbers as representing a rise and each of the negative numbers as representing a fall of the mercury in a thermometer, find the total rise (or fall) in each case; that is, find the algebraic sum of the numbers in each group: 1. 2. 3. 4. 16 12 8 8a 10-4 4 2a 5 6-16 3a 31 il ~± Note The positive sums of Nos. 1, 2, 4, and 5 indicate actual rise; the negative sums of Nos. 3 and 6 indicate actual fall. g 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. e 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. 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 part of the arc intercepted; thus, the angle hlk 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. 66. All the angles about a point together equal four right amrles. 67. Hiscellaneous Review. 1. The angle abd is an angle of about — degrees. The angle dbc is an angle of — degrees. The angle abd + the angle dbc - the angle dbc. The angle dbc is an angle of — degrees. 2. The angle mln is an angle of about — degrees. The angle mln + the angle nls = the angle mis. The angle mis is an angle of about —. 3. 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. |