Mental Work Illustrated. We may omit names of orders. (See In this example 1 ten is taken from 2 tens and changed to 10 units; one of these units is changed to ten tenths. The operation may be represented thus: 39. One concrete number cannot be subtracted from another unless both are expressed in terms of the same unit. For example, we cannot subtract 5 tons from 7 miles; nor can we subtract 3 feet from 60 inches, unless either 3 feet is expressed in inches or 60 inches expressed in feet. 40. It is easily seen that if from a given number several numbers be taken in succession the result will be the same as if the sum of those numbers were subtracted from the given number. Ex. Subtract the sum of 366, 648, and 759 from 2314. 2314 366 648 759 541 9, 8, and 6 make 23; subtract the 3 from the 4 and carry the 2; 2, 5, 4, and 6 make 17; subtract the 7 from 11 and carry the 1; 1, 7, 6, and 3 make 17, which is to be subtracted from 22. 41. When several operations of addition and subtraction have to be performed in succession the result is the same in whatever order the operations are performed. Hence, to find 28 – 15 + 26 — 17 — 14 + 12, first find the sum of 28, 26, and 12, the numbers to be added; then the sum of 15, 17, and 14, the numbers to be subtracted; and finally taking the difference of these two sums; thus, 42. To detect mistakes in subtraction, add the remainder to the subtrahend, and the sum should equal the minuend; or subtract the remainder from the minuend, and the new remainder should equal the subtrahend. EXAMPLES VII. Written Exercises. 1. Subtract 129.6 from 3145, 81.7 from 3002, and 123.4 from 432.1. 2. Subtract 15.97 from 79.15, 18235 from 1000000, and 135.79 from 24680.6. 3. Find the values of 645 - 378, 307 – 149, 294 208, 2179-1984, 3206 — 1679, and 120573 98765. Find the difference between 4. 3.726 and 5.949. 5. 14.753 and 6.876. 6. 1 and .888. 7. .00013 and .00175. 12. Find the values of 8. 3.008 and 3.08. 9. .217 and .271. 10. 20 and .675. 11. .8017 and .00693. (1) 319723+175-184. (2) 151 -77 + 94 – 111. 13. Find 3.17 +4.216 5.8004 +2.0097 - .99873. 14. Find 21.09 -3.985 - 7.0095 + .09372 - 4.38009+ 2.60009. 15. Subtract from 11.214 the sum of 2.301, 1.7293, 2.0507, and 3.62743. 16. Subtract from 20 the sum of 3.416, 2.6008, 5.73124, and 1.5063. 17. Subtract from 121097 the sum of 7916, 1214, 1397, and 34162. 18. Subtract from 1000000 the sum of 421654, 127, 31562, 1795, and 123456. 19. Subtract 27 from 80, and then 27 from the remainder, and so on as many times as possible; and find the final remainder. 20. What number must be taken from 81 to leave 37 as remainder? 21. By how much does the sum of 3.5612 and 4.71305 exceed the sum of 1.70862 and 5.91927 ? 22. What number must be taken from one hundred thousand to leave five thousand four hundred eightyseven as remainder? 23. The difference between two numbers is 145, and the greater is 597; what is the smaller? 24. The sum of two numbers is 1000, and one of them is 594; what is the other? 25. On a man's birthday in 1891 he was 63 years old. In what year was he born? 26. In 1891 a man of 65 was on his birthday just 37 years older than his son. In what year was the son born? 27. Add the sum of 516 and 784 to the difference between 314 and 176. 28. Add the difference between 1925 and 1789 to the difference between 3421 and 1679. 29. In an orchard there are 1572 fruit trees; of these 352 are apple trees, 275 are pear trees, and 187 are plum trees. How many other trees are there? 30. The population of each of five towns is as follows: A, 3789; B, 7861; C, 2893; D, 756; E, 847. If B and D were united, the new town would be how much larger than A, C, and E together? MULTIPLICATION. 43. A short process of adding two or more equal numbers is called Multiplication. = 20. Ex. 1. 5 + 5 + 5 + 5 = 20 ; i.e., 4 fives = 15. If we say (Ex. 1) 5, 10, 15, 20, or (Ex. 2) 3, 6, 9, 12, 15, we are adding by a long process. = 20, or 5 threes = 15, we are adding If we say 4 fives by a short process called multiplication. 44. The number which is to be thus increased is called the Multiplicand. The number which indicates how many equal numbers are to be added is called the Multiplier. The result of multiplication is called the Product. The multiplicand and multiplier are called Factors of the product. 45. The multiplication of any two numbers not greater than nine is easily found by actual addition. It will be shown that every case of multiplication can be reduced to a series of cases of multiplications of numbers not greater than ten; it is therefore essential to learn by heart all the products of such numbers. These products are given in the following table, called the Multiplication Table. 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 11 22 33 44 55 66 24 80 90 100 110 120 77 88 99 110 121 132 4 36 48 60 Any horizontal line in the table gives the products of the number which begins the line by the first twelve numbers in order. Thus the fourth line can be read 1 four is 4, 2 fours are 8, 3 fours are 12, 4 fours are 16, etc. It is usual and desirable, though not absolutely necessary, to learn the Multiplication Table as far as 12 times 12. This table |