Simplify a fraction by dividing numerator and denominator by their common factors
Simplify in steps by dividing by smaller common factors repeatedly
Check whether a fraction is fully simplified (common factors = 1 only)
Simplify fractions in context — probability, test scores, ratios
Real-World Applications
Test scores: You score 18 out of 24. Simplified: \(\frac{18}{24} = \frac{3}{4}\). Much clearer than the raw fraction.
Probability: A bag has 12 red balls out of 20 total. \(P(\text{red}) = \frac{12}{20} = \frac{3}{5}\). Simplified fractions are the standard form in probability.
Ratios: Mixing paint 8 : 12 simplifies to 2 : 3 — the same process as simplifying fractions.
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To simplify a fraction, divide both numerator and denominator by a number that divides into both exactly. Find the largest common factor — the biggest number that divides both — to simplify in one step.
To simplify \(\frac{12}{18}\): List factors → common factors are 1, 2, 3, 6 → largest = 6 → \(\frac{12 \div 6}{18 \div 6} = \frac{2}{3}\)
Example 1 — Simplify \(\frac{12}{18}\)
Step 1
List factors of 12: 1, 2, 3, 4, 6, 12
Step 2
List factors of 18: 1, 2, 3, 6, 9, 18
Step 3
Common factors (in both lists): 1, 2, 3, 6. Largest = 6.
If you cannot immediately spot the largest common factor, divide by any common factor you can see. Repeat until no further simplification is possible. You reach the same answer.
Both are even, so divide by 4: \(\frac{48 \div 4}{72 \div 4} = \frac{12}{18}\)
Step 2
Largest common factor of 12 and 18 is 6: \(\frac{12 \div 6}{18 \div 6} = \frac{2}{3}\)
Step 3
No more common factors. Fully simplified: \(\mathbf{\frac{2}{3}}\)
Example 4 — Simplify \(\frac{36}{60}\)
Step 1
Divide by 4: \(\frac{36 \div 4}{60 \div 4} = \frac{9}{15}\)
Step 2
Divide by 3: \(\frac{9 \div 3}{15 \div 3} = \frac{3}{5}\)
Step 3
No more common factors. Answer: \(\mathbf{\frac{3}{5}}\)
Quick test: Is your fraction fully simplified? If you can divide both numbers by anything greater than 1, keep going.
✓ Quick Check 2 — Simplifying in steps
Question 1 of 10
Simplifying in Context
In real-life problems, write the fraction from the context then simplify. Always verify your answer is fully in lowest terms — numerator and denominator share only 1 as a common factor.
Example 5 — Probability: a bag has 8 blue and 12 red balls. What fraction are blue?
Step 1
Total = 8 + 12 = 20. Fraction blue = \(\frac{8}{20}\).
Step 2
Largest common factor = 4. Simplify: \(\frac{8 \div 4}{20 \div 4} = \frac{2}{5}\)
Step 3
The probability of picking a blue ball = \(\frac{2}{5}\).
Example 6 — Test score: Sarah gets 21 out of 28. Express as a simplified fraction.
Step 1
Score = \(\frac{21}{28}\).
Step 2
Largest common factor = 7. Simplify: \(\frac{21 \div 7}{28 \div 7} = \frac{3}{4}\)
Step 3
Sarah scored \(\frac{3}{4}\) of the marks.
Common mistake: Leaving answers unsimplified. Exam questions almost always expect the fraction in its lowest terms.
✓ Quick Check 3 — Simplifying in context
Question 1 of 10
Practice
Simplify each fraction fully. Write answers as a/b (e.g. 3/4).
#
Question
Your answer
Result
1
Simplify 6/8
2
Simplify 10/15
3
Simplify 12/20
4
Simplify 9/12
5
Simplify 15/25
6
Simplify 8/24
7
Simplify 14/21
8
Simplify 18/24
9
Simplify 16/40
10
Simplify 21/28
⚠ Things to Watch Out For
Dividing by a common factor but not the HIGHEST: 24/36 ÷ 2 = 12/18, then ÷ 3 = 4/6, then ÷ 2 = 2/3. This works but takes longer. Find the HCF (12) to simplify in one step.
Thinking a fraction with small numbers is fully simplified: 4/6 still simplifies to 2/3. Always check whether the numerator and denominator share any common factors.
Applying simplification to addition: You cannot simplify BEFORE adding: (2+4)/(3+6) ≠ 2/3. Simplification only applies to an existing fraction, not to components of a sum.
Dividing numerator and denominator by different numbers: You must divide BOTH by the same number. Dividing top by 4 and bottom by 6 changes the value of the fraction.
Not checking the final answer: After simplifying, check: do the top and bottom share any factors other than 1? If yes, simplify again.