CANCELLATION 120. Since 8 x 5 is contained in 16 × 5 the same number of times that 8 is contained in 16, and since 8 is contained in 16 the same number of times that 1 is contained in 2, it is evident that the factors 5 and 8 may be omitted or canceled from both dividend and divisor without changing the quotient. 121. The process of shortening computations in division by rejecting equal factors from both dividend and divisor is called cancellation. WRITTEN EXERCISES 122. 1. Divide 8 x 77 x 15 by 4 x 44 x 20. 2 7 3 8 × 77 × 15_21 4 × 44 × 20 442 = 25 8 We indicate the division by writing 8 x 77 x 15 above a line and 4 × 44 × 20 below it (§ 85). Since dividing both dividend and divisor by 4 does not change the quotient, the factor 4 is canceled from 8 and 4, leaving 2 in the dividend and 1 (not written) in the divisor. Similarly, 11 is canceled from 77 and 44; 5 from 15 and 20; and 2 from 2, left in the dividend, and from one of the 4's left in the divisor. Since the factors now left in the dividend are prime to those in the divisor, we have 7 × 3 divided by 2 × 4, or 21. The quotient is 2§. NOTE. When all the factors of the dividend are canceled, the resulting dividend is 1. The same is true of the divisor. When all the factors of both dividend and divisor are canceled, the quotient is 1, or 1. Divide, using cancellation: 2. 14 x 27 x 24 x 80 by 32 x 63 x 45 3. 25 x 42 x 18 x 54 by 12 x 70 x 30 x 9 7. Divide .25 × 6.3 × 35 by 4.9 × .12 × 7.5. 3 9 5 5 25 × 63 × 35 × 10 25 49 × 12 × 75 2 7 4 3 2 = 12.5 Omitting decimal points from the dividend multiplies it by 100 × 10, or by 1000. Omitting decimal points from the divisor multiplies it by 10 × 100 × 10, or by 10,000. Consequently, to make both dividend and divisor integral without changing the quotient, after omitting decimal points, we must multiply the dividend by 10. By cancellation the quotient is then found to be 25, or 12.5. 14. 4.5 × .56 × 6.8 × 9.2 by .7 × 9 × 4.6 15. 39 × 144 × 49 × 36 by 156 × 56 × 27 16. 63 × 33 × 51 × 480 by 27 × 280 × 340 17. 7.92 × 2.4 × .035 × 756 by 5.04 × .018 × 800 × .33 18. 432 × 8.4 × .72 × 2.16 × 48.4 by .56 × 19.8 × 57.6 × 16.2 19. 243 × 210 × 147 × 78 × 88 by 162 × 63 × 231 × 32 × 130 20. How much will 7500 bricks cost at $8.40 per 1000? 21. A man gave 12 days' service for 24 bushels of potatoes worth 56 a bushel. How much money did he earn per day? 22. When eggs are worth 32 per dozen, how many dozen eggs will pay for 16 pounds of tea at 40 per pound? 23. A miller bought 14,400 feet of lumber at $20 per 1000 feet, and paid for it in flour at $6 a barrel. How many barrels were required? 24. A teamster carted 4 loads of sand daily for 14 days and was allowed $.90 per load for the work. He took his pay in corn at the market price and received 84 bushels. How much was the corn worth per bushel? FRACTIONS 123. 1. If any one thing is divided into 8 equal parts, what is one part called? What are 3 parts called? 5 parts? 7 parts? 2. One eighth is written; three fourths,-3. Write: three eighths; five eighths; seven-twelfths. 3. In the fraction, what does 6, the number below the line, show? 5, the number above the line? 124. One or more of the equal parts of a unit is called a fraction. 125. The number that shows into how many equal parts the unit is divided is called the denominator. 126. The number that shows how many parts form the fraction is called the numerator. 127. The numerator and denominator of a fraction are called its terms. 128. What are the terms of the fraction? Which is the numerator? the denominator? A fraction is read by reading the numerator (the number of parts) and then the denominator (telling the kind of parts). Thus, is read, "four fifths"; 7 is read, "twenty-seven twentieths." 129. A number expressed by an integer and a fraction is called a mixed number. The mixed number 8 is read, "eight and four fifths"; $4.33 is read, "four dollars, thirty-three and one third cents." 7. Eight hundred four-thousandths. 8. Sixty-four one-hundred-seventy-fifths. 9. Six hundred five four-thousand-eight-hundredths. 10. Forty-five dollars eighty-seven and one half cents. 11. Two hundred twenty-five and ninety-eight one-hundredforty-fourths. REDUCTION 132. The process of changing the form of any number without changing its value is called reduction. 133. Reduction of fractions to higher or lower terms. 1. Draw a line and mark it into halves; then into fourths; then into eighths. 2. How may the terms of be obtained from those of ? from those of ? 3. Which has the larger, or higher, terms, or ? the lower terms, or? the lowest terms, 1, 4, or 1? 134. Multiplying or dividing both terms of a fraction by the same number does not change its value. 135. When the terms of a fraction are prime to each other, the fraction is expressed in its lowest terms. EXERCISES 3; 136. 1. Change to twelfths: ; 1 ; † ; 1 ; † ; §. 12 12 6 4 8 18 16 ;; 1; 13; 10; 11. 8800 36 14 28 626840 24 23 72 24 96 15 2248 From § 134 we know that it will not change the value of to multiply both terms by the same number. Since 96 ÷ 4 = 24, we must multiply both terms by 24. 16 5. to 144ths 8. 11 to 156ths |