Equivalent Fractions

🔵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

👀 See a worked example

Write two fractions equivalent to $\tfrac12$.

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 $\tfrac36$

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

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

Write $\tfrac68$ in its simplest form.

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

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

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

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

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

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

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

🤖 Vidi's 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.

🟢Compare Fractions with Unlike Denominators

Comparing fractions means deciding which one is larger (or whether they are equal). There are three simple cases.

Case 1 — Same denominator. If the bottoms are equal, the fraction with the bigger numerator is bigger: $\tfrac35>\tfrac25$.

Case 2 — Same numerator. If the tops are equal, the fraction with the smaller denominator is bigger — fewer pieces means each piece is larger: $\tfrac14>\tfrac18$.

Case 3 — Unlike denominators. Make the denominators the same first, then compare the tops:

Find a common denominator (usually the LCM of the two denominators).
Rewrite each fraction as an equivalent fraction with that denominator.
Compare the numerators.

Example: compare $\tfrac23$ and $\tfrac34$. The LCM of $3$ and $4$ is $12$:

$$\frac23=\frac{8}{12}\qquad \frac34=\frac{9}{12}\qquad\Rightarrow\qquad \frac34>\frac23$$

Benchmark trick. Compare each fraction to $\tfrac12$. Since $\tfrac38$ is less than $\tfrac12$ and $\tfrac58$ is more than $\tfrac12$, we know $\tfrac58>\tfrac38$ without any calculation.

Symbols: $>$ means greater than, $<$ means less than, and $=$ means equal to.

Common mistake: comparing only the numerators (or only the denominators) when the denominators are different. Always make the denominators the same first.

COMPARING FRACTIONS

Same bottom -> bigger top wins:     3/5  >  2/5

Same top -> smaller bottom wins:    1/4  >  1/8
   (fewer pieces means each piece is bigger)

Different bottoms -> make them equal first:

   2/3 = 8/12        3/4 = 9/12        so  3/4 > 2/3

Benchmark with 1/2:   3/8 < 1/2 < 5/8   so  5/8 > 3/8

👀 See a worked example

Which is greater, $\tfrac35$ or $\tfrac25$?

Same denominator, so compare the tops.
$3>2$
Answer: $\tfrac35$

Which is greater, $\tfrac14$ or $\tfrac18$?

Same numerator, so the smaller denominator is greater.
$4<8$, so $\tfrac14$ has bigger pieces.
Answer: $\tfrac14$

Compare $\tfrac23$ and $\tfrac34$.

LCM of $3$ and $4$ is $12$.
$\tfrac23=\tfrac{8}{12}$ and $\tfrac34=\tfrac{9}{12}$
$9>8$
Answer: $\tfrac34>\tfrac23$

Compare $\tfrac12$ and $\tfrac58$.

Common denominator $8$: $\tfrac12=\tfrac48$.
$\tfrac58$ vs $\tfrac48$: $5>4$
Answer: $\tfrac58>\tfrac12$

Put $\tfrac12$, $\tfrac34$, $\tfrac14$ in order from smallest to largest.

Common denominator $4$: $\tfrac12=\tfrac24$.
So we have $\tfrac24,\ \tfrac34,\ \tfrac14$.
Order the tops: $1<2<3$.
Answer: $\tfrac14,\ \tfrac12,\ \tfrac34$

Use the $\tfrac12$ benchmark to compare $\tfrac38$ and $\tfrac58$.

$\tfrac12=\tfrac48$.
$\tfrac38<\tfrac48$ and $\tfrac58>\tfrac48$.
Answer: $\tfrac58>\tfrac38$

🤖 Vidi's Key Points

Same denominator: the bigger numerator is the bigger fraction.
Same numerator: the smaller denominator is the bigger fraction.
Unlike denominators: make them equal (use the LCM) then compare tops.
Benchmark against $\tfrac12$ to compare quickly.
Symbols: $>$ greater than, $<$ less than, $=$ equal to.
Never compare tops alone when the bottoms are different.

🟣Add and Subtract Fractions with Same Denominator

When two fractions have the same denominator, adding or subtracting them is easy: keep the denominator and just add (or subtract) the numerators.

$$\frac{a}{d}+\frac{b}{d}=\frac{a+b}{d}\qquad\qquad \frac{a}{d}-\frac{b}{d}=\frac{a-b}{d}$$

Why the bottom stays the same: the denominator tells the size of each piece. When we put pieces together or take some away, the size of a piece does not change — only how many we have changes. So only the numerators are added or subtracted.

Examples: $\tfrac15+\tfrac25=\tfrac35$ and $\tfrac45-\tfrac15=\tfrac35$.

Always simplify the answer when you can:

$$\frac26=\frac{2\div2}{6\div2}=\frac13$$

Improper fractions. If the top becomes bigger than the bottom, the answer is more than one whole. For example $\tfrac35+\tfrac45=\tfrac75$, which is the same as $1\tfrac25$.

Common mistake to avoid: do not add the denominators. $\tfrac15+\tfrac25$ is $\tfrac35$, never $\tfrac{3}{10}$.

ADD / SUBTRACT with the SAME denominator:

   keep the bottom, add (or subtract) the tops

     1     2     1 + 2     3
    --- + --- = ------- = ---
     5     5       5       5

     4     1     4 - 1     3
    --- - --- = ------- = ---
     5     5       5       5

Then SIMPLIFY if you can:   2/6 = 1/3

👀 See a worked example

Find $\tfrac15+\tfrac25$.

Same denominator, so add the tops: $1+2=3$.
Keep the bottom: $5$.
Answer: $\tfrac35$

Find $\tfrac45-\tfrac15$.

Subtract the tops: $4-1=3$.
Keep the bottom: $5$.
Answer: $\tfrac35$

Find $\tfrac16+\tfrac16$ and simplify.

$\tfrac16+\tfrac16=\tfrac26$
Simplify: $\tfrac26=\tfrac13$
Answer: $\tfrac13$

Find $\tfrac78-\tfrac38$ and simplify.

$\tfrac78-\tfrac38=\tfrac48$
Simplify: $\tfrac48=\tfrac12$
Answer: $\tfrac12$

Find $\tfrac35+\tfrac45$ and write the answer as a mixed number.

$\tfrac35+\tfrac45=\tfrac75$
$\tfrac75=1\tfrac25$ (since $7\div5=1$ remainder $2$)
Answer: $1\tfrac25$

Riya ate $\tfrac28$ of a pizza and her brother ate $\tfrac38$. How much did they eat together?

$\tfrac28+\tfrac38=\tfrac58$
Answer: $\tfrac58$ of the pizza

🤖 Vidi's Key Points

Same denominator: add or subtract the numerators; keep the denominator.
$\tfrac{a}{d}\pm\tfrac{b}{d}=\tfrac{a\pm b}{d}$.
Never add the denominators — only the numerators change.
Always simplify the answer when possible (e.g. $\tfrac26=\tfrac13$).
If the top becomes bigger than the bottom, write it as a mixed number.
Word problems: "together/total" means add; "left/more than" often means subtract.

🔵Generate Equivalent Fractions

🟢Compare Fractions with Unlike Denominators

🟣Add and Subtract Fractions with Same Denominator

🎯 Mission Test