Example: What would be the cost of ten in. by 6 in. machine bolts at 23. Cancellation.-Very often the work of multiplying fractions may be lessened by cancellation, as it avoids the necessity of reducing the product to lowest terms. To get an idea of cancellation we must first understand what a "factor" is. A Factor of a number is a number which will exactly divide it. Thus, 2 is a factor of 8, 3 is a factor of 27, 5 is a factor of 35, etc. When the same number will exactly divide two or more numbers it is called a common factor of those numbers. Thus, 2 is a common factor of 8 and 12, because it will divide both 8 and 12 without leaving a remainder. 4 is also a common factor of 8 and 12. Similarly, 7 is a common factor of 14 and 21. This idea of common factors we have already used in reducing fractions to lowest terms. Thus, when we have we divide both 8 and 12 by 4 and get 12 Cancellation is a process of shortening the work of reduction by removing or cancelling the equal factors from the numerator and denominator. Example: 2 Suppose we have several fractions to multiply together, as 3 21 X 1432 Their product is 3×2× 3 × 21 378 5376 This is not in its lowest terms so we divide both numerator and denominator by 2, 3, and 7, and get Now, if we had struck out the common factors from the numerator and denominator before multiplying the fractions, we would have shortened the work and our answer would have been in its lowest terms without reducing.. Thus: Explanation: First the 3 in the numerator is cancelled with the 3 in the denominator. This merely divides the numerator and denominator by 3 at the outset instead of waiting until the terms are all multiplied together; and, as 3÷3=1, we cancel a 3 from both numerator and denominator and place I's in their stead. Next we divide both terms by 2. The gives 1 in place of the 2 in the numerator and 2 in place of the 4 in this denominator. Next we see that 7 is a common factor of the numerator and denominator, so we divide the 21 and 14 each by 7 and place 3 and 2 in their places. There are no more common factors; so we multiply together the numbers we now have and get 9 128 Explanation: First we cancel 250 out of 500 and 250; and then 9 out of 36 and 63; then 7 out of 42 and 7; then 10 out of 50 and 20; and finally 2 out of 2 and 2. This removes all the common factors and we get 120 for the 24. Division-The Reverse of Multiplication.-Division is just the opposite of multiplication and this fact gives us the cue to a very simple method of dividing fractions. To divide one fraction by another, invert the divisor and then multiply. To invert means to turn upside down. Invert and we get ; invert and we get 3 4 Explanation: The divisor is Inverting this gives In multiplying, 3 9 8 we make use of cancellation to simplify the work, and we get or the result. 14 16 11 for Suppose we have a fraction to divide by a whole number; as 2 1 ÷2=? 2 is the same as If we invert this we get Therefore, 1 2 25. Compound Fractions. Sometimes we see a fraction which has a fraction for the numerator and another fraction for the denominator. This is called a Compound Fraction. If we remember that a fraction indicates the division of the numerator by the denominator, we will see that a compound fraction can be simplified by performing this division. 27 3 26. How to Analyze Practical Problems.-The chief trouble that students have in working practical problems is in analyzing the problems to find out just what operations they should use to work them. Problems in multiplication or division of fractions will fall in one of the three following cases: 1. Given a whole; to find a part (multiply). 2. Given a part; to find the whole (divide). 3. To find what part one number is of another (divide). Example of Case 1: 7 The total weight of a shipment of steel bars is 3425 lb. of this consists 25 The base of a dynamo weighs 270 lb.; the base is find the total weight. A molder who is on piece work sets up 91 flasks, but the castings from 7 of them are defective. What fractional part of his work does he get paid for? 53. A gallon is about of a cubic foot. If a cubic foot of water weighs 1 15 62 lb., how much does a gallon of water weigh? 54. Aluminum is 23 times as heavy as water; and copper is 8 times as heavy as water. Copper is how many times as heavy as aluminum? 55. If a certain sized steel bar weighs 2 lb. to the foot, how long must a piece be to weigh 81lb.? 56. What is the cost of a casting weighing 3871⁄2 lb. at 41 cents a pound? 57. How many steel pins to finish 3 16 rod if we allow in. to each pin for cutting off and finishing? 58. A certain piece for a machine can be made of steel or of cast iron. 1 If drop forged from steel it would weigh 73 lb. and would cost 63 cents per pound. If made of cast iron. it would have to be made much heavier |