4.4 ADDING and SUBTRACTING LIKE FRACTIONS and LEAST COMMON DENOMINATOR (LCD) Lets say we have a pizza which has been cut into 8 equal slices. Porky ate 4 slices. What fraction of the pizza did he eat? Tiny ate 2 slices. What fraction of the pizza did he eat? All together, how many slices have been eaten? What fraction of the pizza has been eaten? 4 8 = 1 2 2 8 = 1 4 If we make this into a fraction problem, we see we had Or 1 + 1 = ___ 2 4 What do you notice? + ___ = + = _______ In order to ADD or SUBTRACT fractions, they must be written out of the same number of “pieces”, that is the DENOMINATOR must be the SAME. We then ADD the NUMERATORS and keep the DENOMINATORS exactly the SAME. After this, we must REDUCE if possible. Ex: Add the following. You must remember all of your rules for adding signed numbers! a) 1/6 + 2/6 b) -3/10 + 7/10 c) d) -4/15 – 1/15 -7/x3 - -4/ x3 NOTE: In each case, you only combine the NUMERATORS, the DENOMINATORS stay exactly the same until AFTER the operation of addition or subtraction has been done. THEN you may REDUCE the final result (if possible). Suppose the denominators are not “like”. What can we do? Two fractions are EQUIVALENT if they reduce (simplify) to the same fraction: When we SIMPLIFY a fraction, we wish to write its equivalent using the smallest numbers possible. For example, 1/2 would be the simplified form of the fraction 2/4. Sometimes it is possible to simplify a fraction just by performing the operation of division as suggested by the division bar! 15/3 for example, should be written as just 5. Ex: Simplify each of the following -27 17 -9 3 17 1 = = = 0 6 = -6 0 = How would you simplify -3/27? How is this different from –27/3 above? Ex: Are 5/15 and 1/ 3 equivalent? Ex: Are 3/5 and 12/20 equivalent? To write equivalent fractions, we can simply either MULTIPLY or DIVIDE by the SAME NUMBER in BOTH the numerator AND denominator. Note that equivalent fractions will always have the SAME SIGN! a = ac b bc a = ac b bc or Of course, b and c cannot be ______. What we are actually doing is multiplying or dividing by ONE!! Since c/c = 1. (Ex: 5/5 = 1, -2/-2 = 1) 2 4 = 2 4 1 = 2 4 2 = 2 2 2 4 2 = 4 8 Ex: Write an equivalent fraction to 7/9 with a denominator of 63. Ex: Write an equivalent fraction to -3/4 with a denominator of 24. To find a COMMON DENOMINATOR, we find the SMALLEST NUMBER that all existing denominators will DIVIDE INTO EVENLY. We call this denominator the LEAST COMMON DENOMINATOR (LCD). Let’s find a systematic approach to finding the LCD. Case 1: We can see by inspection that there are NO COMMON FACTORS in each denominator. We can then simply multiply the denominators together to find the LCD. Ex: 2 3 3 4 Case 2: The largest of the denominators is divisible by the “smaller” denominators. That largest denominator will then be the least COMMON denominator. Ex: 1 3 5 6 Case 3: There are common factors in the denominators. We can make sure that we do not repeat factors unnecessarily by either “TRIAL AND ERROR”, or by using PRIME FACTORIZATION. Ex: 1 18 5 12 Ex: Find the Least Common Denominator for each of the following: a) -1 3 ; 5 10 b) -2 3 ; 15 10 c) 3 -3 ; 8 10 d) 2 4 ; 3 5 If adding, we then write EQUIVALENT FRACTIONS, and continue with adding the numerators while keeping the denominators the SAME. Do p. 261~ # 36, 39, 44, 74, p 263 Singapore only. To prepare for the next section, add each of the addends on the previous page, and write the sum in reduced form.