Multiplying and Dividing Rational Numbers

Rational Numbers

• • The term Rational Numbers refers to any number that can be written as a fraction.

This includes fractions that are reduced, fractions that can be reduced, mixed numbers, improper fractions, and even integers and whole numbers. • An integer, like 4, can be written as a fraction by putting the number 1 under it.

4  4 1

Multiplying Fractions

• When multiplying fractions, they do NOT need to have a common denominator.

• To multiply two (or more) fractions, multiply across, numerator by numerator and denominator by denominator.

• If the answer can be simplified, then simplify it.

• • Example: 2 5  9 2  2  9 5  2  18 10  2  2  9 5 Example: 3 4  5 2  3  5 4  2  15 8

Cross Cancelling

• • • When multiplying fractions, we can simplify the fractions and also simplify diagonally. This isn’t necessary, but it can make the numbers smaller and keep you from simplifying at the end.

From the last slide: 2 5  9 2  2 5   9 2  18 10 An alternative:

1

2 5  9 2

1

 1  5 9  1  9 5   2 2  9 5 You do not have to simplify diagonally, it is just an option. If you are more comfortable, multiply across and simplify at the end.

Mixed Numbers

• To multiply mixed numbers, convert them to improper fractions first.

  3 2 5     1 1 4      17 5     5 4      17 5

1

    5 4

1

   17  1 1  4  17 4

Try These: Multiply

Multiply the following fractions and mixed numbers: 1) 6 5    1 3   2) 5 1 3  6 5 3)    1 3 4     3 1 2   4) 4 9  6 8

Solutions: Multiply

1) 6 5    1 3    6 15  3  3   2 5 2) 5 1 3  6 5  16  3 6 5  96 15  3  3  32 5 3)    1 3 4     3 1 2     7 4    7 2   49 8 4) 4 9  6 8  24 72  24  24  1 3

Solutions (alternative): Multiply

Note: Problems 1, 2 and 4 could have been simplified before multiplying.

1)

2

6 5    1 3

1

    2 5 2) 5 1 3  6 5  16

1

3  6

2

 5 32 5 4)

1

4 9  6 8

2

 1 9  6 2

1 3

 1 9

3

 3 1

1

 1 3

Dividing Fractions

• When dividing fractions, they do NOT need to have a common denominator.

• To divide two fractions, multiply by the reciprocal…flip it and multiply. 2 5  Change Operation.

9 2  2 5  2 9 Flip 2nd Fraction.

Dividing Fractions

• Finish the problem by following the rules for multiplying fractions.

2 5  9 2  2 5  2 9  4 45

Try These: Divide

• Divide the following fractions & mixed numbers: 1) 6 5 1 2   2)  3 2 1 2   3) 2 1 3  3 2 3 4)  7 3  1 2 3

1) 6 5 2)  3 2

Solutions: Divide

1 2    6 5 1 2    3 2 2 1    12 5 2 1   6  2 2  2  3 1  3 3) 2 1 3  3 2 3  7 3  11 3  7 3  3 11  21  3 33  3  7 11 4)  7 3  1 2 3   7 3  5 3   7 3  3 5   21 15  3  3   7 5