Equivalent Fractions • Topic 1 of 3

Generate Equivalent Fractions

What is a Fraction?

A fraction names a part of a whole. It is written as $\dfrac{\text{numerator}}{\text{denominator}}$. The denominator (bottom number) tells how many equal parts the whole is divided into; the numerator (top number) tells how many of those parts we take. So $\tfrac34$ means $3$ out of $4$ equal parts.

What are Equivalent Fractions?

Equivalent fractions are fractions that look different but name the same amount. For example $\tfrac12$, $\tfrac24$ and $\tfrac48$ all cover exactly half of a bar — they are equivalent.

How to Make an Equivalent Fraction

Multiply or divide the numerator and the denominator by the same non-zero number. Whatever you do to the top, you must do to the bottom:

$$\frac12=\frac{1\times2}{2\times2}=\frac24=\frac{1\times3}{2\times3}=\frac36$$

Simplest Form

A fraction is in simplest form when the top and bottom share no common factor except $1$. To simplify, divide both by their highest common factor. For example $\tfrac68=\tfrac{6\div2}{8\div2}=\tfrac34$.

Fraction	$\times$ what	Equivalent
$\tfrac13$	$\times2$	$\tfrac26$
$\tfrac13$	$\times3$	$\tfrac39$
$\tfrac13$	$\times4$	$\tfrac{4}{12}$

Real life: half a pizza ($\tfrac12$) is the same as two quarter-slices ($\tfrac24$) — equivalent fractions are all around us.

Common mistakes to avoid:

Changing only the top or only the bottom — you must do both.
Adding the same number to top and bottom — that does NOT make an equivalent fraction; you must multiply or divide.
Forgetting to simplify the final answer.

EQUIVALENT FRACTIONS shade the SAME amount of a bar:

  1/2   |#########|---------|        (1 of 2 parts)
  2/4   |####|----|####|----|        (2 of 4 parts)
  4/8   |##|--|##|--|##|--|##|--|     (4 of 8 parts)

All three shade exactly half the bar, so:  1/2 = 2/4 = 4/8

MAKE one by multiplying top AND bottom by the same number:

     1     1 x 2     2          1     1 x 3     3
    --- = ------- = ---        --- = ------- = ---
     2     2 x 2     4          2     2 x 3     6

Solved Examples

Worked Example

Write two fractions equivalent to $\tfrac12$.

Solution

Multiply top and bottom by $2$: $\tfrac{1\times2}{2\times2}=\tfrac24$
Multiply top and bottom by $3$: $\tfrac{1\times3}{2\times3}=\tfrac36$
Answer: $\tfrac24$ and $\tfrac36$

Worked Example

Fill in the missing number: $\tfrac23=\dfrac{\square}{9}$.

Solution

The bottom went from $3$ to $9$, that is $\times3$.
Do the same to the top: $2\times3=6$.
Answer: $\tfrac69$

Worked Example

Write $\tfrac68$ in its simplest form.

Solution

The highest common factor of $6$ and $8$ is $2$.
$\tfrac{6\div2}{8\div2}=\tfrac34$
Answer: $\tfrac34$

Worked Example

Is $\tfrac34$ equivalent to $\tfrac{9}{12}$?

Solution

$\tfrac34=\tfrac{3\times3}{4\times3}=\tfrac{9}{12}$
Yes, they are equal.
Answer: Yes

Worked Example

Simplify $\tfrac{10}{15}$.

Solution

The highest common factor of $10$ and $15$ is $5$.
$\tfrac{10\div5}{15\div5}=\tfrac23$
Answer: $\tfrac23$

Worked Example

Fill in: $\tfrac35=\dfrac{12}{\square}$.

Solution

The top went from $3$ to $12$, that is $\times4$.
Do the same to the bottom: $5\times4=20$.
Answer: $\tfrac{12}{20}$

Key Points

A fraction $\tfrac{a}{b}$ has a numerator (top) and a denominator (bottom).
Equivalent fractions name the same amount, e.g. $\tfrac12=\tfrac24=\tfrac48$.
Make one by multiplying OR dividing top and bottom by the same number.
Never just add a number to top and bottom — multiply or divide.
Simplest form: divide top and bottom by their highest common factor.
Two fractions are equivalent if each simplifies to the same fraction.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.Which fraction is equivalent to $\tfrac12$?

Explanation: $\tfrac24=\tfrac12$.

Q2.To make an equivalent fraction, multiply top and bottom by the:

Explanation: Same number.

Q3.$\tfrac68$ in simplest form is:

Explanation: Divide by $2$.

Q4.Fill in: $\tfrac23=\dfrac{\square}{9}$.

Explanation: $2\times3=6$.

Practice more MCQs → 📝 Assignment · 35 marks