41. The Secretary of the Treasury of the United States estimated the revenue for 1885 to be $330,000,000. The actual revenue was $323,690,706. How much did the actual fall short of the estimated revenue? 42 The population of Chicago in 1860 was 109,260, in 1880, 503,185. Find the increase. 43. The gross earnings of the Eastern Railroad for 1883 were $3,584,506, and the expenses were $2,310,830. Find the net earnings for the year. 44. The population of New York City in 1880 was 1,206,299, in 1860 it was 805,651. Find the increase for twenty years. 45. In 1880 Kentucky raised 149,017,855 pounds of tobacco, and Virginia raised 78,421,860 pounds. How many more pounds did Kentucky raise than Virginia? 46. The value of the tobacco raised in Kentucky in 1880 was $10,431,250, the value of that raised in Virginia was $6,273,749. Find the difference. 47. The population of Massachusetts in 1880 was 1,783,085, and of Virginia 1.512 565. Find the difference. 48. The population of New York in 1880 was 5,082,871, and of Ohio 3,198,062. Find the difference. 49. The population of the United States in 1880 was 50,155,783, in 1870, 38,558,371. Find the increase. 50. In 1885 the railroads of the United States earned from freight $519,690,992, and from passengers $200,883,911. How much more was earned from freight than from passengers? 51. The imports of raw cotton into England in 1871 were 1,778,139,776 pounds, the exports were 362,075,616 pounds. How many more pounds were imported than exported? CHAPTER IV. MULTIPLICATION. 49. If the cost of 6 tons of coal at $7 a ton is required, the amount can be found by writing $7 six times in a column convenient for adding, as in the margin, and finding the sum of the column. 50. If the cost of a whole cargo of coal was required, this operation would be long and tedious, and therefore a shorter process has been devised, called Multiplication. $7 7 7 7 7 7 $42 By this process $7 is written only once, 6 is written beneath the $7 to show the number of times $7 must be taken in order to obtain the required amount, and this amount is found by saying 6 times $7 are $42. Thus: $7 $42 51. In this operation $7 is called the multiplicand, 6 the multiplier, and $42 the product. The multiplier 6 is the sum of six l's, and the product 42 is the sum of six 7's. Hence, it will be seen that: Multiplication is an operation by which, when two numbers are given, called multiplicand and multiplier, a third number is found called product, which is formed from the multiplicand as the multiplier is formed from unity. 52. The multiplicand is the number to be multiplied. The multiplier is the number by which we multiply. The product is the result obtained. The multiplicand and mul tiplier are called factors of the product. The product of two or more factors is the same in whatever order they are taken. Thus, 3 × 4 = 4 x 3. The dots in the margin, read horizontally, make 3 fours; read vertically, make 4 threes. 53. The sign of multiplication is X. When the multiplier precedes the multiplicand, the sign × is read times. Thus, 6×$7=$42 is read 6 times $7 equal $42. 54. When the multiplier follows the multiplicand the sign X is read multiplied by. Thus, $7 × 6=$ 42 is read $7 multiplied by 6 equal $42; and means $7 taken 6 times equal $42. In all cases the product refers to the same kind of units as the multiplicand. 55. Products of two factors, which are each less than ten, must be learned by heart. They can all be readily found by addition. Thus, if the product of 4 times 6 is required, we see that the multiplier 4 is the sum of four 1's, and the multiplicand is 6; hence, the product is the sum of four 6's, and we write In the same way every product is found when each of its two factors is less than ten; and the results are all written in the following multiplication table: Ex. 34. (Oral.) 1. Of what number are 2 and 4 the factors? 3 and 3? 5 and 3? 2 and 5? 3 and 6? 2. What are the factors of 14? of 9? of 8? of 18? of 6? of 21? of 10? 3. 4 is one factor of 8; 3 is one factor of 12; 9 is one factor of 18; what is the other? 4. 3 times what number make 15? 5. 8 times what number make 24? 6 times what number make 24? 6 times what number make 12? 42? 30? 18? 6. 7 times what number make 21? 63? 35? 49? 56? 14? 7. 8 times what number make 32? 64? 16? 40? 24? 56? 8. 9 times what number make 27? 72? 45? 63? 36? 18? |