How do I find equivalent fractions?

Choose a non-zero number k to multiply (or divide) both the numerator and denominator by. For example, to turn \frac{3}{4} into a fraction with denominator 20, calculate 20 \div 4 = 5, then multiply both parts by 5 to get \frac{3 \cdot 5}{4 \cdot 5} = \frac{15}{20}.

Are equivalent fractions always equal in value?

Yes. Equivalent fractions represent the exact same value, even though the numerators and denominators look different. For instance, \frac{1}{2}, \frac{2}{4}, and \frac{50}{100} are all equivalent—they all equal 0.5.

What happens if I multiply the numerator and denominator by different numbers?

You get a fraction with a different value, not an equivalent one. The rule is: multiply (or divide) both by the same non-zero number k. Using different multipliers breaks the equivalence.

Why does the formula say k \neq 0?

If k = 0, you would multiply both numerator and denominator by zero and get \frac{0}{0}, which is undefined. The condition k \neq 0 ensures you always get a valid, meaningful fraction.

Equivalent Fractions

Multiply or divide both parts of a fraction by the same non-zero number to create an equivalent fraction with the same value.

\frac{a}{b} = \frac{ak}{bk}, \quad k \neq 0

Solve a problem with Equivalent Fractions

Type the problem. The solver will use Equivalent Fractions where Equivalent Fractions is the right tool, and tell you when it is not.

What each symbol means

What Equivalent Fractions takes

Symbol	Meaning
$a$	The numerator of the original fraction; the top part of the fraction you start with.
$b$	The denominator of the original fraction; the bottom part of the fraction you start with, which must be non-zero so the fraction is defined.
$k$	The multiplier or divisor; a non-zero number you use to multiply or divide both the numerator and denominator by—it determines how you scale the fraction.

When to use it

Use equivalent fractions when you need to rewrite a fraction with a different denominator but the same value.

Level

Usually taught in: Pre-Algebra

Worked examples

1. Finding an Equivalent Fraction with a Larger Denominator

Problem

Find an equivalent fraction to

\frac{2}{3}

that has a denominator of 9.

$9 \div 3 = 3$
We need to figure out the multiplier. Since the denominator must become 9 and it is currently 3, we divide: $9 \div 3 = 3$ , so $k = 3$ .
$\frac{2}{3} = \frac{2 \cdot 3}{3 \cdot 3}$
Multiply both the numerator and denominator by $k = 3$ following the formula.
$\frac{2}{3} = \frac{6}{9}$
Compute the multiplications: $2 \times 3 = 6$ in the numerator and $3 \times 3 = 9$ in the denominator.

Answer: $\frac{2}{3} = \frac{6}{9}$

This is a forward application of the equivalent fractions rule. We use it when we need a fraction with a specific denominator, such as finding a common denominator for adding fractions later.

2. Simplifying a Fraction with a Negative Numerator

Problem

Write

\frac{-8}{12}

as an equivalent fraction with denominator 3.

$12 \div 3 = 4$
To shrink the denominator from 12 to 3, we determine what divides into 12 to give 3. Dividing $12 \div 3 = 4$ , so we will divide both top and bottom by $k = 4$ .
$\frac{-8}{12} = \frac{-8 \div 4}{12 \div 4}$
Apply the rule by dividing both numerator and denominator by $k = 4$ .
$\frac{-8}{12} = \frac{-2}{3}$
Compute the divisions: $-8 \div 4 = -2$ in the numerator and $12 \div 4 = 3$ in the denominator.

Answer: $\frac{-8}{12} = \frac{-2}{3}$

This example shows that equivalent fractions work with negative numbers, and that dividing both parts by $k$ creates equivalent fractions. This is actually simplifying fractions, which is the reverse operation.

3. Rewriting Lab Measurements as Simpler Fractions

Problem

A lab measures that a liquid's temperature rose by

\frac{120}{300}

degrees Celsius over 10 minutes. Write an equivalent fraction with denominator 5 to express this rate more clearly.

$300 \div 5 = 60$
We want the denominator to become 5. Dividing $300 \div 5 = 60$ , so we will divide both top and bottom by $k = 60$ .
$\frac{120}{300} = \frac{120 \div 60}{300 \div 60}$
Apply the rule by dividing both numerator and denominator by $k = 60$ .
$\frac{120}{300} = \frac{2}{5}$
Compute the divisions: $120 \div 60 = 2$ in the numerator and $300 \div 60 = 5$ in the denominator.

Answer: $\frac{120}{300} = \frac{2}{5}$

In real measurements, fractions often have large numbers that obscure the actual rate. Finding an equivalent fraction with smaller numbers makes the data easier to interpret—here, $\frac{2}{5}$ is much clearer than $\frac{120}{300}$ , yet they represent exactly the same temperature rise.

Common mistakes

Where Equivalent Fractions usually goes wrong

The mistake	Why it is wrong	The fix
$\frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 5} = \frac{8}{15}$	The rule says multiply by the same number $k$ for both numerator and denominator. If you multiply by different numbers, you don't get an equivalent fraction; you get a completely different one with a different value.	Make sure the same number multiplies both top and bottom: $\frac{2}{3} = \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15}$ .
$\frac{2}{3} = \frac{2 \cdot 4}{3} = \frac{8}{3}$	Changing only one part (the numerator or denominator) changes the value of the fraction entirely. $\frac{8}{3}$ is greater than 2, while $\frac{2}{3}$ is less than 1, so they are not equivalent.	Always multiply or divide both the numerator and denominator by $k$ : $\frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}$ .
$\frac{2}{3} = \frac{2 \cdot 0}{3 \cdot 0} = \frac{0}{0}$	The formula explicitly requires $k \neq 0$ . If $k = 0$ , both numerator and denominator become 0, and $\frac{0}{0}$ is undefined and meaningless.	Always choose a non-zero value for $k$ , such as 2, 3, $\frac{1}{2}$ , or any other non-zero number.

Tips and when to use something else

To find the multiplier, divide the new denominator (or numerator) by the old one.
Equivalent fractions always equal the same value—they just look different.
To add or compare fractions with different denominators, use equivalent fractions to create a common denominator; reach for Least Common Multiple to find the best one.
Be careful with negative signs: the fraction $\frac{-2}{3}$ is equivalent to $\frac{2}{-3}$ but not $\frac{2}{3}$ .

Related formulas

Simplifying FractionsSimplify a fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common factor.Adding FractionsAdd fractions with different denominators by finding a common denominator, then combining the numerators to get your answer.Least Common MultipleThe least common multiple is the smallest positive number that all given numbers divide into evenly; use it when adding fractions with different denominators.Comparing FractionsCompare two fractions to find which is larger or smaller without finding a common denominator—use when you need a quick size comparison.

Frequently asked questions

How do I find equivalent fractions?: Choose a non-zero number $k$ to multiply (or divide) both the numerator and denominator by. For example, to turn $\frac{3}{4}$ into a fraction with denominator 20, calculate $20 \div 4 = 5$ , then multiply both parts by 5 to get $\frac{3 \cdot 5}{4 \cdot 5} = \frac{15}{20}$ .
Are equivalent fractions always equal in value?: Yes. Equivalent fractions represent the exact same value, even though the numerators and denominators look different. For instance, $\frac{1}{2}$ , $\frac{2}{4}$ , and $\frac{50}{100}$ are all equivalent—they all equal 0.5.
What happens if I multiply the numerator and denominator by different numbers?: You get a fraction with a different value, not an equivalent one. The rule is: multiply (or divide) both by the same non-zero number $k$ . Using different multipliers breaks the equivalence.
Why does the formula say $k \neq 0$ ?: If $k = 0$ , you would multiply both numerator and denominator by zero and get $\frac{0}{0}$ , which is undefined. The condition $k \neq 0$ ensures you always get a valid, meaningful fraction.

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Reviewed 2026-09-18