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Multiplying a Fraction by a Fraction

If you want to multiply a fraction by a fraction, the best approach is to treat it as two different problems. First, multiply the numerators to get the numerator of the product. Next, multiply the denominators to get the denominator of the product. Put your numerator over the denominator and you have your answer! This can be expressed mathematically like this:

$\frac{a}{b}×\frac{c}{d}=\frac{a×c}{b×d}$

Visualizing multiplying fractions

This concept might get easier if you try working with real numbers, so let's do a practice problem.

$\frac{2}{3}×\frac{5}{7}$

Using the formula above, we have $2×5=10$ for the numerator and $3×7=21$ for the denominator. The final answer is $\frac{10}{21}$ .

You can visualize this by drawing a circle with $\frac{2}{3}$ filled in and then dividing the shaded part into 7 equal parts. Color in five of those 7 pieces and you have a visual representation of $\frac{5}{7}$ ths of $\frac{2}{3}$ :

The relationship between multiplying fractions and simplest form

By now, you're probably accustomed to writing any answer involving fractions in simplest form, and multiplying fractions is no exception. Importantly, just because the fractions you're multiplying are in simplest form doesn't mean your answer will be. Consider the following example:

$\frac{2}{5}×\frac{3}{4}$

Both of these fractions are in simplest form, but $2×3=6$ and $5×4=20$ . The resulting fraction, $\frac{6}{20}$ , is mathematically accurate but not in the simplest form. To rectify that, you need to reduce the fraction by factoring out a 2 from both the numerator and denominator. That gives you a final answer of $\frac{3}{10}$ .

Another useful trick you can try is reducing before you multiply so your original answer is in the simplest form. This would involve expressing the 4 in $\frac{3}{4}$ as $2×2$ and then factoring out one of them along with the 2 in $\frac{2}{5}$ , leaving you with $1×3=3$ for the numerator and $5×2=10$ for the denominator. You get the same final answer in $\frac{3}{10}$ .

This process comes in very handy when we want to simplify radical expressions, so there's nothing wrong with doing things the slow and steady way to make sure you get the right answer.

Multiplying fractions practice problems

a. $\frac{1}{2}×\frac{1}{4}$

To multiply fractions, multiply the numerators together and then multiply the denominators together:

$\frac{1×1}{2×4}=\frac{1}{8}$

The answer is $\frac{1}{8}$ .

b. $\frac{2}{5}×\frac{7}{8}$

$\frac{2×7}{5×8}=\frac{14}{40}$

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2:

$\frac{14}{2}=7$

$\frac{40}{2}=20$

The answer is $\frac{7}{20}$ .

c. $\frac{6}{7}×\frac{2}{3}$

$\frac{6×2}{7×3}=\frac{12}{21}$

Simplify the fraction by dividing both numerator and denominator by their GCD, which is 3:

$\frac{12}{3}=4$

$\frac{21}{3}=7$

The answer is $\frac{4}{7}$ .

d. $\frac{1}{5}×\frac{2}{5}$

$\frac{1×2}{5×5}=\frac{2}{25}$

The answer is $\frac{2}{25}$ .

Get help multiplying fractions with Varsity Tutors today

Multiplying fractions can be a counterintuitive concept since the product is generally lower than the numbers you started with, but it's an important concept that students will need throughout their school years and beyond. If your student tenses up around fractions or needs a jolt of self-confidence to realize their academic goals, a private tutor can help them study in ways that make sense for them. Reach out to the Educational Directors at Varsity Tutors to learn more about the benefits of 1-on-1 instruction.

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