EXERCISES Find the following differences and check results: 1. 375.76 4. 4235.604 7. 4506.325 10. 5672.036 and 2984.236 is 62. Multiplication. The product of 18. This reduces to 0.0161. Hence, the product of 0.7 and 0.023 can be found by multiplying the numbers together and giving this product as many decimal places as those of the multiplicand and multiplier added together. This rule extends to mixed numbers. Multiplication and division of decimals may be checked by casting out the nines just as in whole numbers. Note that this does not check the position of the decimal point. EXERCISES Read the following as a business man or a scientist would and find the products. Check results. 63. Abbreviated Multiplication.-In finding the price of 14.97 T. of coal at $ 12.35, only the last two decimal places are used. Why? Instead of carrying out the multiplication in full and dropping the useless decimal places, the work can be abbreviated and yet be fully as accurate. $12.35 14.97 49.40 (1 1. Multiplying by 4 gives 49.40. 2. Multiplying by 10 gives 123.50. 3. Multiplying by .9 would give 3 decimal places. .9 X .05 gives .045. Instead of carrying 4 we carry 5. If the third decimal had been 4 or less, we should have carried 123.50 (2 only 4. This gives 11.12. A line is drawn through 5 to 11.12 (3 show that it will not enter into the next multiplication. .86 (4 4. Multiplying 3 by .07-since 5 is omitted—gives 184.88 (5 .021, or 2 to carry. Hence, multiplication by .07 gives .86. 5. Adding the numbers as usual gives 184.88. EXERCISES Read the following as a business or scientific man would and find the products to two decimal places: 324)5648.374 324 2408 64. Division.-Division of a decimal by a whole number is carried out quite similarly to that of division of a whole number by another whole number. 17.433 324 goes into 546 tens, 1 ten times. 324 goes into 2,408 units, 7 units times. 2268 1403 324 goes into 1,403 tenths, 4 tenths times. 1296 1077 324 goes into 1,077 hundredths, 3 hundredths times. 972 1054 324 goes into 1,054 thousandths, 3 thousandths times. If the divisor contains a decimal, multiply divisor and dividend each by 10 as many times as is necessary to make the divisor a whole number. This gives the same kind of division as the above. 24.357 ÷ 3.72 2,435.7 ÷ 372. EXERCISES = Read the following, carry out the divisions, and check: 13. 6. 475.386 ÷ 18.45 William works for $7.50 per week. How much is this per day? 14. Mr. Grey bought a lot 66 ft. front for $650. What is this per front foot? Abbreviated Division.-Division of decimals can also 1. 17,348 tens ÷ 4,892 gives 3 tens. 65. be abbreviated. places. 35.464 2. 4892) 173488 14676 1) 26728 24460 2) 2268 1957 3) 311 293 4) 18 19 5) 26,738 units ÷ 4,892 gives 5 units. 3. Instead of annexing zeros to the dividend cut off the last digit from the divisor and divide 2,268 tenths by 489 giving 4 tenths. In multiplying 489 by 4, note that the 2, cut off, times 4 gives 8. Hence 1 is carried to 4 × 9, which is then 36 + 1, or 37. 4. Next cut off 9 from the divisor and divide 311 hundredths by 48, giving 6 hundredths. In multiplying by 6, note that 6 X 9 54, of that 5 is carried to 6 × 8, which is then 48 +5 5. Finally cut off 8 from the divisor and divide 18 thousandths by 4. This is 4 thousandths, since 4 X 8 gives 3 to carry to 4 × 4. = = 53. The divisor must contain one digit more than the number of decimal places needed. A zero must be annexed to the dividend for each such digit lacking in the divisor. In 635 19 to 2 decimal places one zero must be annexed; in 566 to 2 decimal places two zeros must be annexed. Carry out these divisions. EXERCISES Find the following quotients by abbreviated division to 3 decimal places and check your results: 13. Twenty-four dollars were paid for a railroad ticket to travel 723 mi. Find the cost per mile to three decimal places. 66. Reducing Common Fractions to Decimals.—Any fraction can be reduced to a decimal by merely dividing the numerator by the denominator, as means 23 ÷ 42. Thus: 67. Repeating Decimals.-In the above decimal, 476190 keeps on repeating. Such a decimal is called a repeating decimal, a recurring decimal, or a circulating decimal. They are written by placing a dot above the first and last digits of the repeating part. Thus, 0.5476190. 68. Reducing Repeating Decimals to Common Fractions. —It is shown in algebra that a repeating decimal can be reduced to a common fraction by the following rule: Write a fraction with the repeating part as the numerator. Write as many 9's in the denominator as there are digits in the numerator and annex zeros, if necessary, so that with the 9's they equal the number of decimal places of the decimal. 0.0456 456 Thus: 865 9990 = ? and 0.4365 = 10 + 9990 = ? EXERCISES 1. Reduce to decimals, marking the repeating part:, 3, 11, 12, 4, 15, 15, 13, 4, 11, 12, 15, 13. 2 6 3 2. Change to common fractions: 0.425, 0.0264, 4.136, 0.4635, 0.04562, 0.00456, 0.8, 0.045624, 5.4506. 3. The readings on a certain barometer are in inches and fractions of an inch. To compare the readings with the weather maps, the fractions must be changed to decimals. Give the following readings in decimals, without using pencil and paper: 291⁄2 in.; 283 in.; 291 in.; 291⁄2 in.; 283 in.; 29§ in. |