Adding, Subtracting, Multiplying, Dividing FractionsDate: 6/7/96 at 19:29:38 From: Kenneth Andriessen Subject: Fractions Dear Dr. Math, I am ten years old. I don't understand lower-term fractions. I am two months away from fifth grade. I was wondering if you could help me. Date: 6/8/96 at 8:12:19 From: Doctor Anthony Subject: Re: Fractions I am not sure how far you have already got with fractions, but I will assume you are just starting. I will use the notation 1/2, meaning 1 over 2, or 1 divided by 2 to represent the fraction 'a half'. 3/4 means 3 divided by 4 and is 'three quarters'. If I want 'three quarters' of some quantity, say a sum of money, then I can think of dividing the sum into four equal portions, and I will then take three of the portions, leaving one portion behind. So I have 3/4 of the sum and there is 1/4 remaining. The total sum of money is represented by the number 1 in this example, and if we add 3/ 4 to 1/4 we get back to the total 1. If the total sum had been $100 to start with, then I simply multiply by 100 to get the actual portions. So (3/4)*100 = 75, (1/4)*100 = 25 You should note from this example that the bottom line (denominator) gives the number of equal portions into which the total is divided, and the top line (numerator) gives how many of these equal portions you are taking. So 5/8 means divide the total into 8 equal portions and then take 5 of these portions. Adding Fractions: From the example of taking 3/4 and 1/4 of something you saw that if we add these together we get back to the total, 1. Thus 3/4 + 1/4 = 1 This illustrates the rule we use. First you must have the same denominator, (bottom line) in each fraction, then with this as the common denominator, the numerator(top line) is found by simply adding the two top numbers, 3 and 1, in this case. So (3+1)/4 = 4/4 = 1 The rule for subtraction is the same, except that we subtract the numbers on the top line. 3/4 - 1/4 = (3-1)/4 = 2/4 = 1/2 The difficulty arises if the two denominators are not equal. Example: 5/8 - 1/4 Now to make the two denominators equal, we first express 1/4 as 2/8. (This is done by multiplying top and bottom lines by the SAME number, 2. Since we multiplied by the same number we have not changed the value of the fraction.) Subtraction is now 5/8 - 2/8 = (5-2)/8 = 3/8 A slightly more difficult problem arises if one denominator is not a multiple of the other. Example: 5/8 + 2/3 For this problem we make the denominator the smallest number into which both 8 and 3 will divide. This is called finding the LCM (lowest common multiple). For 8 and 3, it is easy to see that 24 is the LCM, and we could write the fractions as 5/8 = 15/24 we multiplied 8 by 3 to get 24, so we must also multiply the top number,5, by 3 to get 15. 2/3 = 16/24 multiply top and bottom by 8. Now we require 15/24 + 16/24 = (15+16)/24 = 31/24 = 1 +7/24 Multiplying Fractions. This is easy. Simply multiply the two top numbers and the two bottom numbers. Example: 2/3 * 5/8 = (2*5)/(3*8) = 10/24 Divide top and bottom by 2. = 5/12 Dividing Fractions. The easy rule to remember is: Turn the SECOND fraction upside down and multiply: Example (2/3)/(5/8) = (2/3)*(8/5) = (2*8)/(3*5) = 16/15 = 1 +1/15 This has given a rather quick run through the basic rules. Write in for more help if there are any particular points you don't understand. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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