PROBLEM SET 32. To find the least common multiple, we need to factor each number and make sure all factors are present in our LCM.

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1 PROBLEM SET To find the least common multiple, we need to factor each number and make sure all factors are present in our LCM. Find the least common multiples: 0,,, Least common multiple 0 00 Least common multiple 7 7 Additionally, we are going to be adding fractions in this section. The English of the fractions tell us what to do when the denominators are the same. For eample, is one-fifth plus two-fifths. This is equivalent to three-thirds. Just like one house plus two houses is three houses. In general: a b a b c c c we keep the denominator the same and add the numerators Of course, with this one, we can reduce: 0 When the denominators are not the same, we need to make them the same by creating equivalent fractions. We create these fractions just like we did in the last problem set by multiplying by an appropriate These fractions do NOT have the same denominator, so they can t be added. But, we can make them have the same denominator. Notice that they are both factors of Thus, we can do the following to change the denominators: Since We can write our problem as: and Page 7

2 Of course, we can reduce this: Here, we want the least common denominator. We could use 0, but 0 would be smaller. What we want is the least common multiple. It s easier to see what that is if we factor each number into primes as done at the start of this section: Page

3 Problem Set Homework A. Put >, < or = in each bo to make the sentence true. ( If necessary, change each fraction to an equivalent fraction with a common denominator.) 7. B. Simplify. (Be sure to fully reduce the answer.) C. Find the least common multiples.,,,,.,, 7, 7. 0a,ab, b,,.,, Page

4 PROBLEM SET This problem set is just more eamples of fraction problems. Simplify First we need to put into improper fractions. And then we need a common denominator of : and 0 So, we have: 0 a Here, the common denominator is a. a a and a a a a So, a a a a a a w 0 w w w Make sure you realize this is just multiplication, so we don t need a common denominator. Page 70

5 7a a a Note that the common denominator will have a factor of a in it. But, if you just pay attention to the numbers (7,, and ), you can see that the common denominator will also have a factor of in it. Thus, the common denominator is a. 7 and 7a a a 7 a Thus, we have: 7a a a a a a a If we just focus on the denominators, we see that the common denominator is Treat these fractions the same as any other: Thus, we have:. If we just focus on the denominators, we see that the common denominator is. Treat these fractions the same as any other: Note that we have to multiply both and by!!!! Thus, we have: 7 7. Here, we have to subtract both the and the!!!! Page 7

6 Problem Set Homework Simplify. y y w w w.. 0. y y 7y a m a m a m k. y y Page 7

7 PROBLEM SETS & Terminology: Recall that if the some of two quantities is zero, they are ADDITIVE INVERSES. If the product of two quantities is one, they are MULTIPLICATIVE INVERSES. Quantity Multiplicative Inverse Additive Inverse 0 w 0 w -0 -w DIVISION is just multiplying by the multiplicative inverse: Page 7

8 Problem Set Homework A. Complete the table. Quantity 7 The quantity s multiplicative inverse. The quantity s additive inverse. 7 a y t r 0 b B. Fill in the boes to make true closed sentences Page 7

9 Problem Set Homework Simplify. 7 m.. m 7. a ab 7.. ab b a 0. y y y z y 0z 0. wt w 0 0 Page 7