Multiplying and Dividing Fractions


Our discussion of fractions and rational expressions in Algebra 1 will begin with multiplying and dividing fractions. This is a pretty easy concept to learn, and isn’t complicated at all. To multiply two fractions, you simply multiply across the top and multiply across the bottom to create the new fraction. For example:

\frac{3}{4} * \frac{2}{5} = \frac{3 * 2}{4 * 5} = \frac{6}{20} = \frac{3}{10} 

To divide fractions, you will turn the division problem into a multiplication problem. You do this by flipping the second fraction and changing the operation to multiplication. For example:

\frac{2}{3} \div \frac{5}{7} = \frac{2}{3} * \frac{7}{5} = \frac{2*7}{3*5} = \frac{14}{15} 

The same idea extends to multiplying rational expressions that include polynomials. For example:

\frac{x+2}{x-3} * \frac{x-6}{x+1} = \frac{(x+2)(x-6)}{(x-3)(x+1)} = \frac{x^2 - 4x - 12}{x^2 - 2x - 3} 

Sometimes it’s necessary to do some cancellations to simplify a fraction. These should be done in the third step of the problem above, which is the next to last step. For example:

\frac{x-1}{x+1} * \frac{x+1}{x-5} = \frac{(x-1)(x+1)}{(x+1)(x-5)} = \frac{x-1}{x-5} 

You can do the same kind of simplification when you’re using fractions that are just numbers. For example:

\frac{4}{3} * \frac{5}{8} = \frac{4*5}{3*8} = \frac{5}{3*4} = \frac{5}{12} 

The multiplication and division of fractions and rational expressions isn’t terribly difficult, but it takes a little bit of practice and a lot of focus to avoid making a lot of mistakes.