Fractions: Equivalent Fractions & Simplifying

Fractions show part of a whole. The bottom number, the denominator, tells how many equal parts the whole is split into. The top number, the numerator, tells how many of those parts you have. Two fractions that name the same amount, like $\frac{1}{2}$ and $\frac{2}{4}$ , are called equivalent fractions.

Before this lesson: You should know how to find the greatest common factor and the least common multiple of two numbers.

Extra practice: The Pre-Algebra Unit 2 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Is a Fraction?

Think of a candy bar broken into 4 equal pieces. If you eat 3 pieces, you ate $\frac{3}{4}$ of the bar.

A bar split into 4 equal parts with 3 parts shaded. The denominator 4 is the number of equal parts, and the numerator 3 is the number of shaded parts. — The whole is split into 4 equal parts, and 3 are shaded, so the bar shows 3/4.

The numerator is the top number. It counts the parts you have.
The denominator is the bottom number. It tells how many equal parts make one whole.
The bar between them means divide. So $\frac{3}{4}$ also means $3 \div 4$ .

Types of fractions

Type	What it means	Example
Proper fraction	The numerator is smaller than the denominator, so the value is less than 1.	$\frac{3}{4}$
Improper fraction	The numerator is equal to or bigger than the denominator, so the value is 1 or more.	$\frac{7}{3}$
Mixed number	A whole number and a proper fraction together.	$2\frac{1}{3}$

Changing between improper fractions and mixed numbers has its own lesson later in this unit.

Equivalent Fractions

Equivalent fractions are fractions that look different but have the same value. Look at these three bars. The shaded part is the same size in each one.

Three bars of the same length. The first is split into 2 parts with 1 shaded, the second into 4 parts with 2 shaded, and the third into 8 parts with 4 shaded. The shaded part is the same length in all three, so 1/2 = 2/4 = 4/8. — The shaded parts line up, so 1/2, 2/4 and 4/8 are equivalent fractions.

To make an equivalent fraction, multiply or divide the top and bottom by the same number. You can use any number except 0.

\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4} \qquad \frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}

This works because multiplying top and bottom by the same number is the same as multiplying by 1. For example, $\frac{2}{2} = 1$ .

How to Simplify Fractions

A fraction is in simplest form when the top and bottom have no common factor except 1. To simplify a fraction:

Find the greatest common factor (GCF) of the numerator and denominator.
Divide the top and the bottom by the GCF.
Check that the only common factor left is 1.

If you don’t see the GCF right away, divide by any common factor. Then repeat until you can’t divide any further.

Worked Examples

Example 1: Write equivalent fractions

Write three fractions that are equivalent to $\frac{2}{3}$ .

\frac{2 \times 2}{3 \times 2} = \frac{4}{6} \qquad \frac{2 \times 3}{3 \times 3} = \frac{6}{9} \qquad \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

Step 1. Multiply the top and bottom by 2, then by 3, then by 4.

Answer: $\frac{4}{6}$ , $\frac{6}{9}$ and $\frac{8}{12}$ are all equal to $\frac{2}{3}$ .

Example 2: Find a missing number

Find the missing number: $\frac{3}{5} = \frac{?}{20}$ .

Step 1. Ask what you multiply 5 by to get 20. Since 20 ÷ 5 = 4, the bottom was multiplied by 4.

Step 2. Multiply the top by the same number: 3 × 4 = 12.

Answer: $\frac{3}{5} = \frac{12}{20}$ .

Example 3: Simplify with the GCF

Simplify $\frac{18}{24}$ .

Step 1. The GCF of 18 and 24 is 6.

\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}

Step 2. Divide the top and bottom by 6.

Answer: $\frac{3}{4}$ . The only common factor of 3 and 4 is 1, so this is the simplest form.

Example 4: Simplify in steps

Simplify $\frac{42}{56}$ .

\frac{42}{56} = \frac{21}{28}

Step 1. Both numbers are even, so divide by 2.

\frac{21}{28} = \frac{3}{4}

Step 2. Both 21 and 28 are in the 7 times table, so divide by 7.

Answer: $\frac{3}{4}$ . You could also divide by the GCF, 14, in one step.

Example 5: Word problem

Maya ate 6 slices of a pizza cut into 8 slices. Leo ate 9 slices of a same-size pizza cut into 12 slices. Did they eat the same amount?

\frac{6}{8} = \frac{3}{4}

Step 1. Maya ate $\frac{6}{8}$ of a pizza. Divide top and bottom by 2.

\frac{9}{12} = \frac{3}{4}

Step 2. Leo ate $\frac{9}{12}$ of a pizza. Divide top and bottom by 3.

Answer: Yes. Both simplify to $\frac{3}{4}$ , so they ate the same amount.

Common Mistakes

Mistake	What it looks like	How to fix it
Adding the same number to top and bottom	$\frac{2}{3} = \frac{3}{4}$	Only multiplying or dividing keeps the value the same.
Stopping before simplest form	$\frac{18}{24} = \frac{9}{12}$	9 and 12 still share 3. Keep going: $\frac{9}{12} = \frac{3}{4}$ .
Changing only the top or the bottom	$\frac{3}{5} = \frac{12}{5}$	Whatever you do to the top, do to the bottom.
Thinking a bigger denominator means a bigger fraction	$\frac{1}{8} \gt \frac{1}{4}$	More parts means smaller parts. $\frac{1}{8}$ is less than $\frac{1}{4}$ .
Putting 0 on the bottom	$\frac{5}{0}$	You can’t split a whole into 0 parts, so the denominator can never be 0.

Comparing Fractions

If two fractions have the same denominator, the one with the bigger numerator is bigger. If not, rewrite them with a common denominator first. The least common multiple of the denominators works best.

For example, compare $\frac{3}{4}$ and $\frac{5}{8}$ . The LCM of 4 and 8 is 8, and $\frac{3}{4} = \frac{6}{8}$ . Since 6 is greater than 5, $\frac{3}{4} \gt \frac{5}{8}$ .

Number line from 0 to 1 split into eighths. Points mark 3/8 and 5/8, and 3/4 is marked at the same spot as 6/8. Fractions farther right are greater. — On a number line, 3/4 is at the same spot as 6/8, and it is to the right of 5/8.

Fractions work like the numbers on a number line: the farther right, the greater the value.

Practice Problems

Find the missing number: $\frac{3}{4} = \frac{?}{12}$
Find the missing number: $\frac{5}{6} = \frac{20}{?}$
Simplify $\frac{10}{15}$ .
Simplify $\frac{16}{40}$ .
Simplify $\frac{27}{36}$ .
Simplify $\frac{45}{60}$ .
Are $\frac{4}{6}$ and $\frac{6}{9}$ equivalent? Explain.
Which is greater, $\frac{2}{3}$ or $\frac{3}{5}$ ?
Put in order from least to greatest: $\frac{1}{2}$ , $\frac{3}{8}$ , $\frac{5}{6}$ .
Sam got 15 of 25 questions right on a quiz. What fraction did he get right, in simplest form?

Show answers

\frac{9}{12}

The bottom was multiplied by 3, so multiply the top by 3.

\frac{20}{24}

The top was multiplied by 4, so multiply the bottom by 4.

\frac{10 \div 5}{15 \div 5} = \frac{2}{3}

The GCF is 5.

\frac{16 \div 8}{40 \div 8} = \frac{2}{5}

The GCF is 8.

\frac{27 \div 9}{36 \div 9} = \frac{3}{4}

The GCF is 9.

\frac{45 \div 15}{60 \div 15} = \frac{3}{4}

The GCF is 15.

Yes. Both simplify to $\frac{2}{3}$ .

Use the common denominator 15: $\frac{2}{3} = \frac{10}{15}$ and $\frac{3}{5} = \frac{9}{15}$ . So $\frac{2}{3}$ is greater.

Use the common denominator 24: $\frac{12}{24}$ , $\frac{9}{24}$ and $\frac{20}{24}$ . From least to greatest: $\frac{3}{8}$ , $\frac{1}{2}$ , $\frac{5}{6}$ .

10.

He got $\frac{15}{25}$ right. The GCF is 5, so that is $\frac{3}{5}$ .

Want more practice? The Pre-Algebra Unit 2 Practice Pack has 58 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Fractions FAQ

What is the simplest form of a fraction?

A fraction is in simplest form when the numerator and denominator share no factor except 1. For example, $\frac{3}{4}$ is in simplest form, but $\frac{6}{8}$ is not.

Can the denominator of a fraction be 0?

No. The denominator tells how many parts the whole is split into, and you can’t divide by 0.

Is a fraction like 6/6 equal to 1?

Yes. When the top and bottom are the same (and not 0), the fraction equals 1. Six sixths make one whole.

What is .375 as a fraction?

$0.375 = \frac{375}{1000}$ . The GCF of 375 and 1000 is 125, so it simplifies to $\frac{3}{8}$ .

Summary

A fraction shows part of a whole: the numerator over the denominator.
Equivalent fractions have the same value. Multiply or divide the top and bottom by the same number to make one.
To simplify, divide the top and bottom by their GCF.
To compare fractions, give them a common denominator first.
The denominator can never be 0.

Previous lesson: Least Common Multiple

Next lesson: Adding and Subtracting Fractions

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