# Multiplying fractions

Multiplying fractions can be done following a few relatively simple steps. Unlike adding or subtracting fractions, we do not need a common denominator. We can immediately multiply any two or more fractions following these rules:

- Multiply all the numerators of each fraction being multiplied
- Multiply all the denominators of each fraction being multiplied (the order of steps 1 and 2 can be reversed)
- Write the product of the numerators and denominators in the numerator and denominator of the new fraction, respectively
- Simplify the result if necessary

Examples

Solve:

First we multiply the numerators:

2 × 4 = 8

Next, multiply the denominators:

5 × 7 = 35

So,

The numbers 8 and 35 don't share any factors, so the fraction is already simplified.

In this next example, we will need to simplify:

The above fraction is not yet simplified because 10 and 54 share the factor 2. So we divide 10 by 2 and 54 by 2 to get:

They are equivalent fractions.

### Multiplying fractions and whole numbers

The process for multiplying fractions and whole numbers is mostly the same. We just need to write a whole number as a fraction to multiply it. A whole number in fraction form can be represented by what is known as an improper fraction. Simply, an improper fraction is a fraction where the value of the fraction is greater than 1.

To represent a whole number in fraction form, we can simply treat the whole number as the numerator of the fraction while putting a 1 in the denominator, since 5 ÷ 1 is still 5. It is the same number, but allows us to see the integer 5 as a fraction.

Examples

Solve:

We first rewrite 12 as a whole number, then multiply the fractions:

Once you are comfortable with whole numbers and fractions, it is not necessary to write the whole number in fraction form. 1 multiplied by anything in the denominator will keep the denominator the same, so we just need to multiply the whole number by the numerator, then simplify the fraction.

### Multiplying mixed fractions

Multiplying mixed fractions mainly just requires that we convert the mixed fraction into an improper fraction before multiplying.

Example

Solve:

First we look at the mixed number, . To convert this into an improper fraction, we multiply the denominator, 4, by 2, and then add the numerator. This gives us the numerator of the improper fraction, while the denominator of the improper fraction stays the same. So:

2 × 4 + 3 = 11, so

To understand why, we can look at this as a fraction addition problem. We know that we need a common denominator to be able to add fractions. The number 2, in equivalent fractions is . Another way we could look at this is that 2 = 1 + 1, and 1 with a common denominator is equivalent to . Regardless how we represent the 2 in fractions, when we add it to the we get:

which is what we got when we converted using the method described above.

Now we can finish the multiplication problem:

This is already simplified, but if we wanted to, we could also represent it in mixed fractions by reversing the steps shown above.

77 divides 36 twice, leaving a remainder of 5, so: