LESSON: Multiplying Fractions

READ: Multiply Fractions and Mixed Numbers

Multiply Fractions and Mixed Numbers

You have already learned how to add and subtract fractions, but when you have a fraction and you want to figure out a part of that fraction, you need to multiply. Remember, that a fraction is a part of a whole. Sometimes it is tricky to figure out when to multiply fractions when you are faced with a real-world problem. First, let’s learn how to actually multiply fractions and then we can look at applying this to some real-world problems.

Multiplying fractions is always at least a two-step process.

First, you line up two fractions next two each other, and then you are ready to start multiplying.

$\frac{1}{2} \cdot \frac{4}{5}$

Notice that we used a dot to show that we were multiplying.

You will multiply twice. First, multiply the numerators and write the product of the numerators above a fraction bar. Next, multiply the denominators and write that product underneath the fraction bar. You don’t have to find a common denominator. You do, however, have to reduce your answer to simplest terms. We usually think of multiplying as increasing, but don’t be surprised to get a product that is smaller than one of the factors that you are multiplying.

Example

$\frac{1}{2} \cdot \frac{4}{5}=\frac{1 \times 4}{2 \times 5}=\frac{4}{10}$

Now we have a fraction called $\frac{4}{10}$ . What next?

We can simplify the fraction four-tenths, but dividing the top and the bottom number by the greatest common factor. The greatest common factor of four and ten is two. We divide the numerator and the denominator by two.

$\frac{4}{10}=\frac{4 \div 2}{10 \div 2}=\frac{2}{5}$

Our final answer is $\frac{2}{5}$ .

What about a fraction and a whole number?

When you multiply a fraction and a whole number, we have to make the whole number into a fraction. Then you can multiply across just as you normally would with two fractions and simplify your answer if possible.

Example

$5 \cdot \frac{1}{2}= \frac{5}{1} \cdot \frac{1}{2}=\frac{5}{2}=2 \frac{1}{2}$

How do we multiply mixed numbers?

Because mixed numbers involve wholes and parts, multiplying mixed numbers requires an extra step. Remember improper fractions? It’s essential that you convert mixed numbers to improper fractions before you multiply. Once you have the mixed numbers in the improper fraction form, multiply the numerators together and then multiply the denominators together. If you have an improper fraction as your product, you can convert it back to a mixed number to write your final answer.

Example

$3 \frac{1}{2} \cdot 2 \frac{1}{3}$

First, we convert each to an improper fraction.

$3 \frac{1}{2} &= \frac{7}{2}\\ 2 \frac{1}{3} &= \frac{7}{3}$

Next, we multiply the two improper fractions.

$\frac{7}{2} \cdot \frac{7}{3}=\frac{49}{6}$

Now we can convert this improper fraction to a mixed number.

$\frac{49}{6}=8 \frac{1}{6}$

Our final answer is $8 \frac{1}{6}$ .

Sometimes, when you multiply fractions or mixed numbers, you can end up with very large numbers. When this happens, you can simplify BEFORE multiplying. You simplify on the diagonals by using the greatest common factor of the numbers on the diagonals.

Example

$\frac{2}{9} \cdot \frac{18}{30}$

If we look at the numbers on the diagonals, we can see that there are greatest common factors both ways. The greatest common factor of two and thirty is 2. We can divide both by two to simplify them. The greatest common factor of 9 and 18 is 9. We can divide both by 9. Let’s simplify on the diagonals now.

$\xcancel{\frac{2}{9} \cdot \frac{18}{30}} = \frac{1}{1} \cdot \frac{2}{15}$

Now we multiply across for our final answer.

The answer is $\frac{2}{15}$