the same number, or divided by the same number, the quotient remains unchanged. NOTE.-Division is the inverse of multiplication; therefore, the associative and commutative laws hold in division. (See 55.) 70. To test the accuracy of division: 1. Review the work. 2. Apply principle 3. 3. Cast out the 9's. One example will be given as an illustration of test 3. EXAMPLE 71. Divide and test: 5864 ÷ 86. SOLUTION: 5864 ÷ 86 = 68, with remainder 16. Now, 86 × 68 + 16 = 5864. Excess in 86=5 SHORT PROCESSES IN DIVISION 72. RULE.-To divide by some power of 10: Cut off as many figures in the dividend as there are zeros in the divisor. The figures cut off will be the remainder, and the others the quotient. Example. Divide 34793 by 100. SOLUTION:-34793 ÷ 100 = 347, with remainder 93. 73. To divide by a composite number. Example.-Divide 1728 by 36. Since 36=3×3×2×2, we may divide by the factors; 74. To divide by an aliquot part of 10, 100, 1000, etc. To divide by 21, 12, 16, 25, 33, etc., is the same as to divide by 10, 100, 100, 100, 100, etc. RULE. To divide by an aliquot part of 10, 100, 1000, etc.: Multiply by the denominator and cut off as many figures from the right as there are zeros in the numerator. Example. Divide 58200 by 121. SOLUTION.-12 = 180; 58200+ 100 = 1 of 8 × 58200 = 4656. 75. Approximations in Division. In a problem involving large decimals, an approximate answer is usually all that is desired, and in some cases all that is possible. Much time and labor may be saved by using the following method. Example. Divide 2.614746 by 1.123456, correct to the second decimal place. OPERATION. 1.123456)2.614746(2.32 2.247 (1 carried) 367 337 (1 carried) 30 22 EXPLANATION.-Since only three figures are required in the answer, all but four may be neglected in each, the dividend and divisor. The figures marked will not affect the answer required, if proper allowance be made for the effect they may have upon the partial product. Observe that 1 is added to the 6, making 7 in the first partial product. After each division one figure of the divisor is neglected, but its effect upon the partial product is kept in mind. A TEST FOR THE FOUR FUNDAMENTAL OPERATIONS. 76. The following is a very simple test of the accuracy of addition, subtraction, multiplication, and division. In fact, it is a modification of the method of casting out the 9's. It may be called the unitate method. The unitate of a number is the sum of its digits reduced to units order. Note these examples: The unitate of 32798 = 29 = 11= 2. The sum of the digits of the first number is 29; these digits added=11, and these added 2. The unitate of 32798 2. Note the following example in multiplication: = = 6 x 848 = 12 = 3. = 484 726 242 The unitate in 31944 = 21 = EXPLANATION.-If the work is correct, the unitate of the product equals the unitate of the 3. product of the unitates of the multiplicand and multiplier. It is not necessary to write down as many figures as are written above. The unitate of each number can easily be found mentally. Try this method with the following: 1. Add 37864, 33977, 47693, 36821. 2. Subtract 34689 from 78601. 3. Multiply 46031 by 527; 59993 by 587. PRECEDENCE OF SIGNS 77. A number with no sign preceding it is considered positive. The signs take effect in the following order: 1. The signs of addition and subtraction take effect in the order in which they occur. Thus, 2 + 3−4 +7-2=6. 2. The signs of multiplication and division take effect in the order in which they occur. Thus, 24 ÷ 6X2=8, not 2. NOTE.-Authorities differ on this point. 3. The signs of multiplication and division both take effect before the signs of addition and subtraction. Thus, 36+2X3-8÷4x2=36+6-4-38. 4. The above statements must be modified when braces, brackets, parentheses, or vinculums are used. The expressions in the braces, brackets, parentheses, and vinculums must be simplified first. Thus, 1. 24 × (9 + 6) ÷ (28 – 18 × 2) = 24 × 15 ÷ 20 = 18. 2. 6× {4-[3+(5−3+2)]} = 6× { 4–[3+(5−5)] } . NOTE. The distributive law for multiplication and division may be illustrated by the following: 4 (6 − 2) = 4 × 6 − 4 × 2, and Exercise V Divide and test by casting out the 9's: 1. 1554768 by 216; 1674918 by 189. 2. 65980064 by 5004; 47863 by 7525. Find the quotient, correct to the second decimal place: 3. 58.140625÷7.625; 100 ÷ 3.14159. 4. 997.21567897 ÷ 37.7565936. Find the quotient, correct to the nearest 1000: 5. 225000000÷46.55; 93000000 ÷ 245. 6. In a certain year the revenue of the United States Government was $403080983, which was $6.577 to each person; what was the population in that year? 8. (105÷21) + (80 ÷ 5 ×81) + (36 ÷ 9). 9. 36 [30 ÷ (3 + 5 +7) + 24] + 10 ÷ 5. 10. 96 ÷ 12 × 2 + 15 ÷ 5 – 4 × 2. 11. 16×4÷8−7+48÷16-3-7×4×0×9×16+24×6 +48-4X9÷12. |