Equivalent Fractions • Topic 2 of 3

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:

  1. Find a common denominator (usually the LCM of the two denominators).
  2. Rewrite each fraction as an equivalent fraction with that denominator.
  3. 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
Solved Examples
1
Worked Example

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

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

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

Solution
  • Same numerator, so the smaller denominator is greater.
  • $4<8$, so $\tfrac14$ has bigger pieces.
  • Answer: $\tfrac14$
3
Worked Example

Compare $\tfrac23$ and $\tfrac34$.

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

Compare $\tfrac12$ and $\tfrac58$.

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

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

Solution
  • Common denominator $4$: $\tfrac12=\tfrac24$.
  • So we have $\tfrac24,\ \tfrac34,\ \tfrac14$.
  • Order the tops: $1<2<3$.
  • Answer: $\tfrac14,\ \tfrac12,\ \tfrac34$
6
Worked Example

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

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

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.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.With the same denominator, the bigger fraction has the bigger:
Explanation: Bigger top.
Q2.With the same numerator, the bigger fraction has the ___ denominator.
Explanation: Smaller bottom = bigger pieces.
Q3.Which is greater, $\tfrac23$ or $\tfrac34$?
Explanation: $\tfrac{8}{12}<\tfrac{9}{12}$.
Q4.To compare unlike fractions, first make the ___ equal.
Explanation: Common denominator.
Practice more MCQs → 📝 Assignment · 35 marks