I. Cancelling applied to Compound Fractions. RULE 1. If there be numbers in the numerators and denominators, that be alike, an equal number of the same value may be cancelled. 1. Reduce of of of of to a simple fraction. In this question, we find a 3, 4, and 8 among the numerators, and also the same numbers among the denominators. These we cancel before we commence the operation. 2. What is the value of of † of } of {} ? 7 We find in this question, 8, 11, and 17 among the numerators, also the same numbers among the denominators. These we cancel. of of 13 of 7 of 1 of $25. 3. What is the value of 7× 8 ×13× 7 ×15×25 1225 8X13X15X10X17X 1 170 = $7 Ans. 4. Reduce of 1 of 1 of 1 of 43 to a simple fraction. 5 XXXX12×17×19 5 = 14 Ans. 5. Required the value of off of 18 of of 40. 焉 7×9×10×13×40 9×10×13×24 280 = 113 Ans. 6. Reduce of 1 of 2 to its equivalent value. 7. What is the value of of 1 of 1 of 33 of $ 18? 8. What is the value of of of of $72? RULE 2. When there are any two numbers, one in the numerators, and the other in the denominators, which may be divided by a number without a remainder, the quotients arising from such division may be used in the operation of the question, instead of the original numbers. The quotients also may be cancelled, as other numbers. the 2 above the 14, and 1 below the 7. We also find a 21 among the numerators, and a 27 among the denominators, which may be divided by 3, and that their quotients will be 7 and 9. We write the 7 above the 21, and 9 below the 27. We again find a 5 among the numerators, and a 25 among the denominators, which may be divided by 5, and that their quotients will be 1 and 5. We write the 1 over the 5, and the 5 below the 25. We then multiply the 4, 2, 7, and 1 together for a numerator = 56, and the 1, 9, 5, and 11 for a denominator = 495. The answer will therefore be 56 495. 2. Reduce of off of to a simple fraction. 3. What is the value of of of 1 of 1 of $34 ? 1 3 2 2 4× 9 ×15×14×34 7×20×16×17× 1 1 4 4 1 NOTE. The above rule will apply, when the product of several numbers is to be divided by the product of other numbers. 4. What is the continued product of 8, 4, 9, 2, 12, 16, and 5 divided by the continued product of 40, 6, 6, 3, 8, 4, and 20? The product of 4 and 9 in the upper line is equal to the product of 6 and 6 in the lower, therefore they are cancelled; and the product of 2 and 12 in the upper line is equal to the product of 3 and 8 in the lower line; also the product of 16 and 5 in the upper line is equal to the product of 4 and 20 in the lower line; these are all cancelled. We also find, that the 8 in the upper line and the 40 in the lower line may be divided by 8, and their quotients will be 1 and 5. We write the 1 above the 8 and the 5 below the 40. By the usual process, we now find our answer is }. 5. What is the continued product of 12, 13, 14, 15, 16, 18, 20, 21, and 24, divided by the continued product of 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11? 3 2 3 2 2 2 7 2 12×13×14×15×16×18×20×21×24 2×3×4×5×6×7×8×9×10×11 1 1 1 1 1 1 1 1 1 II. In finding the common multiple of two or more numbers, any one number that will measure another may be cancelled. 1. What is the least common multiple of 4, 6, 8, 12, 16, 10, and 20? 4) 4 6 8 12 16 18 20 3 4 4X3X4X5=240 Ans. 5 By examining this question, we find that 8 may be divided by 4, 12 by 6, 16 by 8, and 20 by 10; therefore we cancel 4, 6, 8, and 10. 2. What is the least common multiple of 5, 15, 30, 7, 14, and 28? 2) 5 x5 30 7 14 28 15 14 2X15X14 420 Ans. = In this question, we find that 15 may be measured by 5, 30 by 15, 14 by 7, and 28 by 14; we therefore cancel 5, 15, 7, and 14. 3. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8, and 9? 2) 1 2 3 4 5 6 7 8 9 2×3×5×7×4×3=2520 3) 5 3749 5 17 4 3 [Ans. 4. What is the least common múltiple of 9, 8, 12, 18, 24, 36, and 72? 9 8 12 18 24 36 72 72 Ans. 5. What is the least number that 18, 24, 36, 12, 6, 20, and 48 will measure? 4) 18 24 36 12 6 20 48 4×3×3×5×4=720 Ans. RULE. When the first and second terms, or the first and third terms, can be divided by any number without a remainder, their quotients may be used in the operation of the questions instead of the terms themselves. |