What you need to know

Things to remember:

  • To simplify a fraction, we find common factors of the numerator and denominator.
  • We divide by the common factor to simplify the fraction.
  • To fully simplify a fraction, we just divide by the highest (biggest) common factor.

We need to remember that factors can be thought of in two ways:

  • Factors are two whole numbers we multiply to make another number.

4\times8=32

4 and 8 are factors of 32

  • We can also do the opposite, division. If we get a whole number when dividing, then these are factors.

32\div2=16

2 and 16 are factors of 32

We also need to remember that common factors are just factors that are the same for two numbers. Let’s do an example.

What are the common factors of 12 and 18?

Factors of 12: 1, 2, 3, 4, 6, and 12

Factors of 18: 1, 2, 3, 6, 9, and 18

Common factors of 12 and 18 are: 1, 2, 3, and 6.

 

But now that we’ve found them, what do we do with them? To simplify a fraction, we divide the top and bottom by the common factors.

Simplify \dfrac{12}{18}.

We simplify in two steps.

Step 1: Find the common factors of the numbers in the fraction.

We’ve already done this!

 

Step 2:

Divide the by the common factors.

 

\frac{12}{18}=\frac{12\div1}{18\div1}=\frac{12}{18}

\frac{12}{18}=\frac{12\div2}{18\div2}=\frac{6}{9}

\frac{12}{18}=\frac{12\div3}{18\div3}=\frac{4}{6}

\frac{12}{18}=\frac{12\div6}{18\div6}=\frac{2}{3}

So, we’ve simplified it, but do we need to do all of these? Not usually, no! Let’s have a look at them.

\frac{12}{18}=\frac{12\div1}{18\div1}=\frac{12}{18}

Didn’t change the fraction, so we can ignore this one.

\frac{12}{18}=\frac{12\div2}{18\div2}=\frac{6}{9}

\frac{12}{18}=\frac{12\div3}{18\div3}=\frac{4}{6}

\frac{12}{18}=\frac{12\div6}{18\div6}=\frac{2}{3}

With these, each time we could divide by a bigger number until we divided by 6. So, really, we just need to divide by the highest (biggest) common factor! When we divide by the highest (biggest) common factor, we call this “simplifying fully”.

 Fully simplify \frac{20}{24}.

Step 1: Find the common factors of the numbers in the fraction.

Factors of 20: 1, 2, 4, 5, 10, and 20

Factors of 24: 1, 2, 4, 6, 12, and 24

Common factors of 20 and 24: 1, 2, and 4.

Step 2: Divide by the highest (biggest) common factor.

\frac{20}{24}=\frac{20\div4}{24\div4}=\frac{5}{6}

Example Questions

Question 1: Simplify \frac{32}{48}

Answer

Step 1: Find the common factors of the numbers in the fraction.

 

Factors of 32: 1, 2, 4, 8, 16, and 32.

 

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48

 

Common factors of 20 and 24: 1, 2, 4, 8 and 16.

Step 2: Divide by the common factors.

\frac{32}{48}=\frac{32\div1}{48\div1}=\frac{32}{58}

 

\frac{32}{48}=\frac{32\div2}{48\div2}=\frac{16}{24}

 

\frac{32}{48}=\frac{32\div4}{48\div4}=\frac{8}{12}

 

\frac{32}{48}=\frac{32\div8}{48\div8}=\frac{4}{6}

 

\frac{32}{48}=\frac{32\div16}{48\div16}=\frac{2}{3}

Question 2: Fully simplify \frac{12}{36}.

Answer

Step 1: Find the common factors of the numbers in the fraction.

 

Factors of 12: 1, 2, 3, 4, 6, and 12

 

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36

 

Common factors of 12 and 36: 1, 2, 3, 4, 6, and 12.

Step 2: Divide by the highest (biggest) common factor.

 

\frac{12}{36}=\frac{12\div12}{36\div12}=\frac{1}{3}