Mastering Fraction Addition: Techniques and Examples for Common Denominators

1. Adding FractionsAbsent copy12/6,7

2. 1st Method to adding fractions Cross Multiply both denominators in order to get a common denominator. (small numbers) 3 + 3 5 4 Multiply by 4 to 3/54 • 3 + 3 • 5Multiply by 5 to 3/4 4 • 5 + 4 • 5 12 + 15 20 is the common denominator 20 20 Add the numerators12 + 15 = 27 = 1 7/20 20 20

3. Example 1 • Solve: write answer in simplest form 1 + 2 2 5 5 · 1 + 2 · 2 5 · 2 5 · 2 5 + 4 1010 5 + 4 = 9 Solution • Are these denominators both small numbers? • Yes they are both small. • What should we do to get denominators the same? • Because they are small we can cross multiply. • After we cross multiply to get the denominators the same then what? • We add the numerators and keep the denominator • Do we have to reduce this fraction? • No we don’t 9 10

4. Method 3 to adding fractions We can use prime factorization to find the common denominator by multiplying the highest number of factors in each denominator. 1 + 5 8 12 Factor each denominator 8 12 2 • 2 • 2 3 • 2 • 2 Multiply the highest # of factors2 • 2 • 2 • 3 = 24Denominator Multiply the right factor to get denominator3 • 1 + 5 • 2 3 • 8 12 • 2 Add the numerators3 + 10 = 13 2424

5. Example 2 • Solve: write answer in simplest form -5 + -68 72 90 3 · 3 · 2 · 2 · 2 3 · 3 · 2 · 5 3 · 3 · 2 · 2 · 2 · 5 = 360 5 · -5 + -68 · 4 5 · 72 90 · 4 -25 + -272 = -297 360 360 360 Solution • What do we do to find the common denominator by using prime factorization? • We factor each denominator. • What prime factors do we see the most of in each denominator? • There is three 2’s and two 3’s with denom72 and one 5 with denom. 90. • What do we do with those factors to find the common denominator? • We multiply the factors together. • What do we do to each fraction in order to get the common denominator? • We find the right factor that we can multiply each fraction with to get the common denominator of 360. • After we get a common denominator what do we do? • We add the numerators and keep the common denominator. Simplify if needed. -297 360

6. Example 3 • Evaluate the expression When a = 1 b = 1 4 3 a + b 1 + 1 4 3 3 · 1 + 1 · 4 3 4 3 4 3 + 4 = 7 12 12 12 Solution • What do we do first? • We substitute the fractions for the variables. • What method do you think would work the best in finding the common denom? • They are small numbers so we should cross multiply to find common denom. • What do you do first? • We cross multiply each fraction to the opposite fraction. • What do we do second? • We add up the numerators and keep the denominator. • What do we have to check before we are finished? • We should check to see if the fraction should be simplified. 7 12

7. Example 4 • Solve: write in simplest form. 9 + 1 14 77 2 · 7 7 · 11 2 · 7 · 11 = 154 11 · 9 + 1 · 14 11 · 14 77 · 14 99 + 14 = 113 154 154 154 Solution • What do we do to find the common denominator by using prime factorization? • We factor each denominator. • What prime factors do we see the most of in each denominator? • There is one 2 and one 7 with denom14 and one 11 with denom. 77. • What do we do with those factors to find the common denominator? • We multiply the factors together. • What do we do to each fraction in order to get the common denominator? • We find the right factor that we can multiply each fraction with to get the common denominator of 360. • After we get a common denominator what do we do? • We add the numerators and keep the common denominator. Simplify if needed. 113 154

8. Example 5 • Simplify: 8 + 2 a b b · 8 + 2 · a b a b a 8b + 2a ab Solution • What method do you think would work the best in finding the common denom? • To cross multiply • What do you do first? • Cross multiply the denominators • Can we add the numerator? Yes or No • What is the proper way to write the numerator? • In alphabetical order 2a + 8b ab