41. Find the sum of 4.327, 8.29, and .836. Operation. Explanation. 4.327 The sum of the units of the third decimal order is 13 ; 8.29 this is equal to 1 unit of the second decimal order and .836 3 units of the third decimal order. Write the 3 units of the third decimal order and add the 1 unit of the 13.453 second decimal order to the other units of that order. The sum of the units of the second decimal order is 15; this is equal to 1 unit of the first decimal order and 5 units of the second decimal order. Write the 5 units of the second decimal order and add the 1 unit of the first decimal order to the other units of that order. The sum of the units of the first decimal order is 14; this is equal to 1 unit of the first integral order and 4 units of the first decimal order. Write the 4 units of the first decim:il order and add the 1 unit of the first integral order to the other units of that order. The sum of the units of the first integral order is 13; this is equal to 1 unit of the second integral order and 3 units of the first integral order, each of which is written in its place. The sum of 4.327, 8.29, and .836 is 13.453. 42. PROBLEMS. 6. Add 85.997, 47.9994, 72, and 53.93. 7. Find the sum of two hundred and six thousandths, and two hundred six thousandths. 8. Find the sum of seven hundred ninety-eight and nine hundred ninety-four thousandths, and seven hundred ninetyfour thousandths. (a) Find the sum of the eight sums. TO THE PUPIL. Can you solve these eight problems and find the sum of the eight sums on first trial without an error ? Addition - United States Money. 43. Find the sum of the money represented in the following columns : $324.45 Explanation. 28.47 Sums of the units of each order : 375.28 6.94 Second decimal order (cents) 81 175.89 First decimal order (dimes) 70 82 56 802.21 Third integral order 39 354.48 Observe that the 8 of the first sum is included in the 916.37 70 of the second sum; that the 7 of the second sum is 144.50 included in the 82 of the third sum ; that the 8 of the 75.34 third sum is included in the 56 of the fourth sum, and 8.88 that the 5 of the fourth sum is included in the 39 of the 246.25 fifth sum. Hence, the sum of the five sums will be $3962.01 represented by the figures 396201. . 44. PROBLEMS. * 1. 3. 146 854 321.75 281 719 286.44 675 325 156.82 284 716 244.31 552 448 986.24 496 504 275.46 285 715 383.27 628 372 152.10 397 603 (a) Find the sum of the four sums. 4. 851.76 13.76 724.54 616.73 847.90 (P. 319.) * TO THE PUPIL. — Remember that nothing short of absolute accuracy is of any value in such work as this. Addition. - Denominate Numbers. 45. Find the sum of 7 bu. 2 pk. 5 qt., 3 bu. 3 pk. 3 qt., 6 bu. 1 pk. 7 qt., and 9 bu. 3 pk. 5 qt. Operation. Explanation. 7 bu. 2 pk. 5 qt. The suin of the number of quarts is 20; 3 bu. 3 pk. 3 qt. this is equal to 2 pecks and 4 quarts. Write the 4 quarts and add the 2 pecks to the pecks 6 bu. 1 pk. 7 qt. given in the second column. 9 bu. 3 pk. 5 qt. The sum of the number of pecks is 11; 27 bu. 3 pk. 4 qt. 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. 46. PROBLEMS. 1. Add. 2. Add. 6 bu. 2 pk. 6 qt. 5 bu. O pk. 7 qt. 4 bu. 2 pk. 2 qt. 4 bu. 1 pk. 6 qt. 5 bu. 2 pk. 2 qt. 3 bu. O pk. 1 qt. 6 bu. 3 pk. 7 qt. 5 bu. O pk. 5 qt. 4 bu. 3 pk. 3 qt. 1 bu. 1 pk. 2 qt. 8 bu. 2 pk. 6 qt. 2 bu. 3 pk. 3 qt. 7 bu. O pk. 5 qt. 4 bu. 2 pk. 4 qt. 5 bu. 1 pk. 4 qt. 2 bu. O pk. 6 qt. 8 bu. O pk. 6 qt. 6 bu. O pk. 5 qt. 7 bu. 3 pk. 2 qt. 1 bu. 3 pk. 2 qt. 3 bu. 3 pk. 3 qt. 2 bu. 2 pk. O qt. 6 bu. 1 pk. 7 qt. 7 bu. O pk. 2 qt. 7 bu. 2 pk. O qt. 3 bu. 2 pk. 2 qt. 2 bu. 3 pk. 6 qt. 3 bu. 1 pk. 2 qt. 6 bu. 1 pk. 6 qt. 3 bu. 2 pk. 1 qt. (P. 320.) (a) Find the sum of the two sums. Algebraic Addition. 47. A coefficient is a number that indicates how many times a literal quantity* is to be taken ; thus, in the expression 4 ab, 4 is the coefficient of ab. When no coefficient is expressed, it is understood that 1 is the coefficient; thus, in the expression 4 a + b, the coefficient of b 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 + abc. There are only two terms in the following: 8 a x 4b +5 a + 6 6. 49. Positive terms are preceded by the plus sign. 50. Vegative 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. 5x+3x+2x= 6. 4 ab+ 2 ab+3 ab= 53. LANGUAGE EXERCISE. Referring to the problems given above, use the following words in complete sentences : Coefficient. Terms. Positive. Similar. Literal. The word quantity in algebra means number. The expression literal quantity means number expressed by letters. + The term coefficient 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 coeficient. Algebraic Addition, 54. Regarding the following positive numbers as representing gains and the negative numbers as representing losses, fic. the total gain (or loss) in each case; that is, find the algebraic sum of the numbers in each group : 1. 2. 3. 4. 5. 6. 96 2 ab -36 - 6 ab 95 50 - 20 8 a За 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. 5. 6. 16 12 8 76 5c - 36 16 - 12c 8 a 2 a Зc 3 a rise; NOTE. — The positive sums of Nos. 1, 2, 4, and 5 indicate actual the negative sums of Nos. 3 and 6 indicate actual fall. |