32÷8=4, 16÷8=2; i. e., dividing the dividend by 2 divides the quotient by 2. ABT. 73. If the dividend remains the same, multiplying the divisor divides the quotient; thus, 24 ÷ 2 = 12, . e., multiplying the divisor by 3 divides the quotient by 3 ART. 74. If the dividend remains the same, dividing the divisor is multiplying the quotient; thus, 369 = 4, e., dividing the divisor by 3 multiplies the quotient by 3. ART. 75. Since multiplying the dividend multiplies the quotient (Art. 71), and multiplying the divisor divides the quotient (Art. 73), it follows, that multiplying both dividend and divisor by the same number does not alter the quotient. ART. 76. Since dividing the dividend divides the quotient (Art. 72), and dividing the divisor multiplies the quotient (Art. 74), it follows, that if the dividend and divisor are both divided by the same number the quotient is not altered. CANCELLATION. ARTICLE 77 Ex. 1. What is the quotient of 12 X 8 X 5 × 3 divided by 8 × 3 × 2? Analysis. Since the factors 8 and 3 are common both to the divisor and the dividend, we may strike out or cancel these two factors from each, since it is equivalent to dividing both dividend and divisor by these factors (Art. 76); we then have 12 X 5 to be divided by 2, or 60 divided by 2, giving the quotient 30. Ans. 30. 2. Divide the product of 18, 10 and 5 by the product of 6, 5 and 2. Analysis. The factor 5, being common to both divisor and dividend, may be canceled. Since 6 in the divisor is a factor. of 18 in the dividend, we may divide 18 by 6, canceling these two numbers, and substituting 3 in place of them; and since 2 in the divisor is a factor of 10 in the dividend, we may cancel 10 and 2, and put 5 in their place. The product of the remaining factors, 3 and 5, will be the quotient. Ans. 15. The above operations are called CANCELLATION. Hence, ART. 78. CANCELLATION is the method of shortening arithmetical operations by striking out equal factors from the divisor and the dividend. From the Analysis we derive the following RULE. Cancel all the factors common to the dividend and divisor; then divide the product of the remaining factors of the divi dend by the remaining factors of the divisor NOTE 1.-When a factor is entirely canceled, it does not leave 0, but the quotient 1 takes its place; because any quantity divided by itself gives 1 for a quotient. Hence, for every factor entirely canceled, either in the dividend or in the divisor, the factor 1 will remain. NOTE 2.-It should be remembered that one factor in the divisor can cancel only one equal factor in the dividend. EXAMPLES. 3. Divide the product of 4, 3, 9 and 7 by the product of 10, 4, 15 and 7. Ans. 4. Divide the product of 63, 105, 20 and 17 by the product of 15, 30, 21 and 7. Ans. 34. 5. What is the quotient of 18 × 36 × 29, divided by 27 X 24? Ans. 29. 6. Divide the product of 12, 32, 5 and 20 by the product. of 16, 24 and 8. Ans. 12. Ans. 30. 7. What is the quotient of 7 X 42 × 5 × 72, divided by 21 X 14 X 12? 8. What is the quotient of 10 X 27 X 9 X 20 × 19, divided by 15 X 18 X 12 X 15? Ans. 19. Ans. 7. 9. What is the quotient of 3, 5, 7 and 20, divided by 5 × 6 x 10? 10. Divide the product of 6, 5, 7, 8 and 66 by the product of 7, 8, 9 and 11. 11. Divide the product of 8, 7, 12, 15 and 17 by the product of 3, 4, 5, 14 and 4. Ans. 51. Ans. 20. 12. How many pieces of cloth, each containing 36 yards, at 15 cents a yard, would be required to pay for 6 boxes of tobacco, each weighing 72 pounds, at 30 cents a pound? Analysis. The value of the tobacco will evidently be equa. OPERATION. 2 2 6 × 72 × 30 =24 pieces to 6 X 72 X 30; and the value of one piece of cloth will equal 36 X 15; hence, dividing the whole value of the tobacco by the value of 1 piece of cloth, the quotient must be the number of pieces. Ans. 24 pieces. 13. How many pounds of sugar, at 16 cents a pound, would pay for 19 pounds of coffee, at 32 cents per pound? Ans. 38 pounds. 14. How many boxes of soda, each weighing 24 pounds, at 12 cents a pound, would be required to pay for 8 sacks of rice, each containing 84 pounds, at 9 cents per pound? Ans. 21 boxes. NOTE. Cancellation may be applied to Fractions, Proportion and various other Rules of Arithmetic. It teaches no new principles, but simply serves to shorten the mechanical opera tions in working examples. PROPERTIES AND RELATIONS OF NUMBERS. ARTICLE 79. One number is said to be a DIVISOR or MEASURE of another when the former is contained in the latter without a remainder. Thus, 6 is a divisor of 12; 5 is a divisor of 15. Name a divisor of 21; of 30; of 42; of 45; of 72; of 84. An EVEN number can be divided by 2 without a remainder; 2, 4, 6, etc. are even. An ODD number cannot be divided by 2 without a remainder; thus 1, 3, 5, 7, etc. are odd numbers. Give all the even numbers up to 40. Give all the oda numbers to 43. ART. 80. All numbers are either prime or composite. A PRIME number is one that can be divided only by itself and 1. Thus, 2, 3, 5, 7 and 11 are prime numbers. A COMPOSITE number is one that can be divided by some whole number besides itself and 1. Thus, 12 is a composite number, because it can be divided by 2, 3, 4 and 6. Mention all the prime numbers up to 43. Give all the composite numbers up to 50. EXAMPLES. 3. Divide the product of 4, 3, 9 and 7 by the product of 10, 4, 15 and 7. Ans. Ans. 34. 4. Divide the product of 63, 105, 20 and 17 by the product of 15, 30, 21 and 7. 5. What is the quotient of 18 × 36 × 29, divided by 27 X 24? Ans. 29. Ans. 30. 6. Divide the product of 12, 32, 5 and 20 by the product of 16, 24 and 8. Ans. 12. 7. What is the quotient of 7 × 42 × 5 × 72, divided by 21 × 14 × 12? 8. What is the quotient of 10 × 27 × 9 × 20 × 19, divided by 15 X 18 X 12 × 15? Ans. 19. 9. What is the quotient of 3, 5, 7 and 20, divided by 5 × 6 x 10? 10. Divide the product of 6, 5, 7, 8 and 66 by the product of 7, 8, 9 and 11. 11. Divide the product of 8, 7, 12, 15 and 17 by the product of 3, 4, 5, 14 and 4. Ans. 51. Ans. 7. Ans. 20. 12. How many pieces of cloth, each containing 36 yards, at 15 cents a yard, would be required to pay for 6 boxes of tobacco, each weighing 72 pounds, at 30 cents a pound? Analysis. The value of the OPERATION. tobacco will evidently be equa 2 2 to 6 X 72 X 30; and the value 6 × 72 × 30 =24 pieces of one piece of cloth will equal 36 X 15; hence, dividing the whole value of the tobacco by the value of 1 piece of cloth, the quotient must be the number of pieces. Ans. 24 pieces. 13. How many pounds of sugar, at 16 cents a pound, would pay for 19 pounds of coffee, at 32 cents per pound? Ans. 38 pounds. |