In this explainer, we will learn how to find the product of two proper fractions by multiplying the numerators and the denominators and write the answer in simplest form.

Let us recall quickly what a fraction actually is. A fraction compares a part to a whole and describes what we call a proportion. The denominator of the fraction is the number of equal shares (or “portions”) the whole is split into, while the numerator is the number of these shares that make the part we are considering.

Now, we can of course perform operations on fractions. We can add a quarter to a half , which gives us three-quarters , or decide that one-tenth of the cake is not a big enough slice and take two slices instead (i.e., ). In the last example, we have multiplied a fraction by an integer. Here, we are interested in multiplying a fraction by another fraction. This can be understood as the fraction of the fraction of a given whole.

Consider this situation: Two people each eat for lunch one-third of the leftover of a pizza made the previous day. The leftover is two-fifths of the whole pizza. What fraction of the pizza are they going to eat together? This fraction of the pizza is illustrated in the diagram.

To find what fraction of the pizza this is, we need to split the leftover ( of the whole) in 3 equal shares and take two of them. As we can see in the next diagram, splitting the in 3 gives .

Note that dividing a fraction (here ) by 3 is equivalent to multiplying its denominator by 3. And a portion of two is simply .

Like a fraction of a given number is given by multiplying the fraction by a number, the fraction of a fraction is equivalent to a multiplication of fractions. So, of of the whole is given by

We can also envision the product of fractions as an area, considering that both fractions describe lengths (as fractions of a length unit). The previous example of can then be represented as the shaded area inside a unit square (i.e., of side 1 and area 1) in the diagram shown.

The dimensions of the grey rectangle are and , and so its area is given by . We see that the unit square is split in 15 equal parts, each having an area of . The grey rectangle is made of 4 of them. Therefore, its area is . We have again illustrated the fact that

Note that if the square were our pizza that we had first cut in 5 equal pieces (pink lines), a portion of two-thirds of two-fifths would indeed be represented by the shaded area.

We have seen that the result of multiplying fractions is obtained by simply multiplying the numerators together and the denominators together:

This square representation clearly shows why: the product of the denominators gives the number of equal shares the square has been divided into, and the product of the numerators gives indeed the number of shares that make the shaded area.

Once we know that, multiplying fractions is quite straightforward. However, we often want to express a fraction in its simplest form. And to avoid doing double work, it is more efficient, when multiplying fractions, to look for common factors between the numerator and the denominator rather than multiply the numerators together and the denominators together and only then reduce the fraction.

Let us see in the first example how this works.

### Example 1: Multiplying Simple Fractions

Calculate , giving your answer in its simplest form.

### Answer

When we multiply fractions together, the result is a fraction where the numerator is the product of all the numerators of the original fractions, and the denominator is the product of all the denominators of the original fractions. So, here we have

Now, we need to check whether there are some common factors between the numerator and the denominator . We see that 2 is a common factor of 2 in the numerator and 4 in the denominator. We can rewrite our fraction as and we see that it can be simplified by 2:

Now we can carry out the multiplication in the denominator, which leads to

Let us look at another example involving three fractions.

### Example 2: Multiplying Three Fractions

Calculate , giving your answer as a fraction in its simplest form.

### Answer

When we multiply fractions together, the result is a fraction where the numerator is the product of all the numerators of the original fractions, and the denominator is the product of all the denominators of the original fractions. So, here we have

Now, we need to check whether there are some common factors between the numerator and the denominator .

We see that 21 and 7 have a common factor ; 3 and 12 have a common factor ; and 11 and 44 have a common factor . Therefore, we can rewrite our fraction as which can be simplified by 3, 7, and 11:

The numbers have been written as a sum of factors here to show exactly what happens. If you are familiar with the method, you can simply cross out, for instance, 21 and 7 and write 3 in the numerator as the result of the simplification by 7.

Now, we can carry out the multiplication in the denominator, which leads to

We are going to look at an example involving mixed numbers.

### Example 3: Multiplying Mixed Numbers

Calculate .

### Answer

Here, we are asked to multiply mixed numbers. The easiest way here is to convert the mixed numbers to fractions, multiply the fractions, and convert the result back to a mixed number.

Both mixed numbers given here have their fractional parts as thirds, so we are going to convert them into thirds.

We find that and

We now need to multiply these two fractions together:

And, finally, we convert this fraction back to a mixed number. There are 6 times 9 in 55, with a remainder of 1, so we get The answer is .

The next example involves three mixed numbers.

### Example 4: Multiplying Three Mixed Numbers

Calculate .

### Answer

We are asked here to multiply mixed numbers. The easiest way here is to convert the mixed numbers to fractions, multiply the fractions, and convert the result back to a mixed number.

We find that

We need now to multiply these three fractions together:

Before multiplying the numerators together and the denominators together, we check for common factors between the numerators and the denominators. We find that 3 and 15 will simplify with 9, as well as 16 with 2 and 4:

The answer is 10.

The last example is a word problem.

### Example 5: Multiplying Mixed Numbers in a Word Problem

Find the volume of a wooden box that measures feet.

### Answer

We are given the three dimensions of a wooden box. We know that the shape of a box is a cuboid. Hence, to find its volume, we need to multiply its three dimensions together. Since the dimensions in feet are given as mixed numbers, we are going, first, to convert the mixed numbers to improper fractions.

We find that

We now need to multiply these three fractions together:

Before multiplying the numerators together and the denominators together, we check for common factors between the numerators and the denominators. We find that 25 will simplify with 5:

By carrying out the multiplication, we find . Finally, we need to convert it back to a mixed number. There are 28 times 16 in 455, with a remainder of 7; therefore,

The answer is that the volume of the wooden box is
ft^{3}.

### Key Points

- A fraction compares a part to a whole. The denominator of the fraction is the number of equal shares (or “portions”) the whole is split into, while the numerator is the number of these shares that make the part we are considering.
- The fraction of another fraction of a given whole is found by multiplying a fraction by the other fraction.
- The product of fractions can also be interpreted as an area, considering that both fractions describe lengths (as fractions of a length unit). For instance, can be represented as the shaded area inside a unit square (i.e., of side 1 and area 1) in the diagram shown.