CHAPTER III MULTIPLICATION AND DIVISION OF FRACTIONS 19. A Whole Number Times a Fraction. In the study of multiplication, we learn that multiplying is only a short way of adding. 4X7 is the same as four 7's added together. Either 4x7, or 7+7+7+7 will give 28. If we apply this same principle to the multiplying of fractions, we see that 4X is the same as four of these fractions added together. This shows that multiplying a fraction by a whole number is performed by multiplying the numerator by the whole number and placing the product over the denominator of the fraction. In other words, the size of the parts is not changed, but the number of parts is increased by the multiplication. After multiplying, the product should be reduced to lowest terms and, if an improper fraction, should be reduced to a whole or mixed number. Example: What would be the total weight of 12 brass castings each weighing of a pound? 20. "Of" Means "Times."-The word of" is often seen in problems in fractions, as for instance, "What is of 5 in.?" In such a case, we work the problem by multiplying, so we say that "of" means "times." You can see that this is so by taking a piece of wood 5 in. long and cutting it into four equal parts and then taking three of these parts. These three parts will be of 5 in., and by actual measurement will be 3 in long, so we know that of 5-33. Now see what times 5 is: which is the same value. Therefore, we see that the word "of" in such a case signifies multiplication. 21. A Fraction Times a Fraction. To multiply two or more fractions together, multiply the numerators together for the numerator of the product and multiply the denominators together for the denominator of the product. Explanation: The numerator of the product is obtained from multiplying the numerators together: 7X2-14. The denominator of the product, in the same manner, is 8X3-24. This 14 gives the product 24' which can be reduced to 7 12 Let us see what multiplication of fractions really means, and why the work is done as just shown. Suppose we are to find of in. This means that of an inch is to be divided into 4 equal parts and 3 of these parts are wanted. If we divide in. into 4 equal parts, each part will be one-fourth as large as in. and, therefore, can be considered as being made up of 7 parts, each one-fourth as large as in. Then of 72. Three of these parts will naturally contain three times as many thirty-seconds, or 3. Therefore: = 22. Multiplying Mixed Numbers.-This is one of the most difficult operations in the study of fractions, unless one adopts a fixed rule and follows it in all cases. The student will have no trouble if he will first reduce the mixed numbers to improper fractions, and then multiply these like any other fractions. To multiply a mixed number by a whole number, we can reduce the mixed number to an improper fraction and then multiply it; or we can multiply the fractional part and the whole number part separately by the number and then add the products. Example: 1 What would be the cost of ten in. by 6 in. machine bolts at 1 30 or 36, or 33 Set this down. Then multiply 30 4 10 by 1. This gives 10, and we add this to the 3 13 cents, Answer. giving a total of 13° 4 3 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 ΤΣ Cancellation is a process of shortening the work of reduction by removing or cancelling the equal factors from the numerator and denominator. Example: Suppose we have several fractions to multiply together, as 3..2 3 21 ·X. 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 1'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. 9 8 or 1 for we make use of cancellation to simplify the work, and we get the result. Suppose we have a fraction to divide by a whole number; as 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. |