Simplifying means writing a fraction in its lowest form — the same value, but with smaller, cleaner numbers. Like tidying your desk — nothing changes, it just looks much neater!
Key IdeaEquivalent fractions look different but are equal
💡 Both bars are identical in length. 6/8 and 3/4 hold exactly the same amount — the pieces are just bigger in 3/4. Simplifying just uses fewer, larger pieces.
🎛️ Build a fraction — drag the sliders
6
8
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Simplified: 3/4
Why Simplify?3 good reasons
Clearer to read — "3/4 of a tank" is much easier than "18/24 of a tank"
Fewer errors — small numbers are simpler to multiply and divide
Exams require it — T Level questions say "give your answer in its simplest form" and marks are lost if you don't
📌 In engineering, gear ratios, mix ratios and tolerances are always in simplified form. 3:4 is much more useful on a blueprint than 36:48!
What counts as "simplest form"?
A fraction is in its simplest form when no whole number (other than 1) divides evenly into both the top and bottom.
Ask: does any number other than 1 divide into BOTH?
YES → not done yet. Keep simplifying.
NO → finished ✅ simplest form!
✅ 3/4 — only 1 goes into both 3 and 4. Done!
❌ 6/8 — 2 goes into both. Not done yet!
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See It Visually
Watch the bars get fewer and wider as we simplify — but always stay the same total length.
Visual ProofSimplifying 8/12 — watch the bars stay the same length!
🔑 All three bars start and end at exactly the same point (shown by the red dashed lines). The shaded area never changes — only the size of the individual pieces does. That's what "equivalent fractions" means!
Circle Model8/12 = 4/6 = 2/3 — same slice, different number of cuts
🍕 Think of a pizza — cutting it into 12 slices and eating 8, or cutting into 3 slices and eating 2, gives you the same amount of pizza. 2/3 is just the simplest way to describe it.
▶ Step-through: simplify 8/12 one step at a time
Start — 8/12
8 out of 12 pieces filled. HCF(8,12) = 4 — we can simplify!
÷ 2 → 4/6
Same total length — 6 bigger pieces now. HCF(4,6) = 2 — still not done!
÷ 2 again → 2/3 ✅
✅ Simplest form! HCF(2,3) = 1. All three bars are the same width.
Step 1 of 3
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Finding the HCF
The HCF (Highest Common Factor) is the biggest whole number that divides exactly into both the top and bottom numbers. Find it, divide both by it — done in one step!
🔍 Interactive HCF Finder — try any two numbers
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What is HCF?Explained simply
Factors are all whole numbers that divide into a number exactly (no remainder).
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors (in both): 1, 2, 3, 6
HCF = 6 ← the HIGHEST one
💡 Divide both by the HCF to simplify in one step: 12/18 ÷ 6/6 = 2/3 ✅
Three MethodsPick the one that suits you
① List factors (best for small numbers)
Write out ALL factors of each, spot the biggest match
② Keep halving (quick when both are even)
Both even → ÷2. Keep going till one is odd. Check for 3, 5 etc.
③ Prime factors (T Level depth)
Break each number into primes, multiply the shared ones
Factor GridFull factor table for 24 and 36
Number
÷ 1
÷ 2
÷ 3
÷ 4
÷ 6
÷ 8
÷ 9
÷ 12
÷ 18
÷ 24
÷ 36
24
✅
✅
✅
✅
✅
✅
❌
✅ HCF!
❌
✅
❌
36
✅
✅
✅
✅
✅
❌
✅
✅ HCF!
✅
❌
✅
✅ HCF(24, 36) = 12. So: 24/36 ÷ 12 = 2/3 in one step.
⚡ Live Simplify — enter any fraction
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Step-by-Step Method
Follow this process every time. Two methods — use whichever is faster for the numbers in front of you.
Method A — HCFSimplify in ONE step (most efficient)
Write out all factors of the numerator (top number)
Write out all factors of the denominator (bottom number)
Identify the HCF — the largest number in both lists
Divide BOTH numerator and denominator by the HCF
Check: HCF of the new fraction = 1? If yes → done ✅
Method B — Keep HalvingGreat when both numbers are even
Are both numbers even? → divide both by 2
Still both even? → divide by 2 again. Repeat.
When one becomes odd → stop halving. Try dividing by 3, 5, 7…
When nothing divides both evenly → you're in simplest form ✅
⚠️ Never divide just one number. You must always divide BOTH top AND bottom by the same amount, otherwise the fraction's value changes and your answer is wrong.
✅ Final check: after simplifying, confirm the HCF of your new top and bottom = 1. If it is, you're done. If not, keep going!
✏️
Worked Examples
Four fully explained examples — easy to engineering challenge level.
Example 1 — Simplify 18/24
Factors of 18:1, 2, 3, 6, 9, 18
Factors of 24:1, 2, 3, 4, 6, 8, 12, 24
HCF:6 — highest number in both lists
Divide top:18 ÷ 6 = 3
Divide bottom:24 ÷ 6 = 4
Answer:18/24 = 3/4 ✅
1/6
Example 2 — Simplify 45/60
Check even?:45 is ODD — can't halve. Try other factors.
Try ÷ 5:45÷5=9, 60÷5=12 → 9/12
Try ÷ 3:9÷3=3, 12÷3=4 → 3/4
Check HCF:HCF(3,4)=1 ✅ done!
Answer:45/60 = 3/4 ✅
1/5
Example 3 ⚙️ Gear ratio 36/48
Context:Drive gear = 36 teeth. Output = 48 teeth.
Factors of 36:1,2,3,4,6,9,12,18,36
Factors of 48:1,2,3,4,6,8,12,16,24,48
HCF:12
Simplify:36÷12=3, 48÷12=4
Answer:3/4 ✅ — for every 4 drive turns, output turns 3 times
1/6
Example 4 — Concrete mix 15/25
Context:15 kg cement in 25 kg batch. What fraction is cement?
HCF(15,25):= 5
Simplify:15÷5=3, 25÷5=5
Answer:Cement = 3/5 of the mix ✅
1/4
⚠️
Mistakes to Avoid
These cost marks in the T Level exam. Recognise them now so you never make them.
Mistake 1Dividing only ONE side
❌ 12/20 → divide top only → 3/20 The value changes completely — 3/20 ≠ 12/20
✅ Always divide BOTH: 12÷4=3 and 20÷4=5 → 3/5
Think of it like a see-saw — what you do to one side, you MUST do to the other to keep it balanced.
Mistake 2Stopping too early
❌ 24/36 ÷ 2 = 12/18 → "done" 12/18 can still be simplified — you stopped too soon!
✅ Always verify: HCF(12,18)=6 → still not done. 24/36 → 12/18 → 6/9 → 2/3 ✅
Always check the final fraction — HCF should be 1 before you stop.
Mistake 3Dividing by a non-factor
❌ 9/15 ÷ 2 = 4.5/7.5 2 doesn't divide into 9 — you get decimals, which is wrong!
✅ Always check the number divides EXACTLY. 9/15: try 3 → 9÷3=3, 15÷3=5 → 3/5 ✅
Mistake 4n/n = 0 (wrong!)
❌ 8/8 → "top minus bottom = 0" This is completely wrong!
✅ Any fraction where top = bottom = 1 8/8 = 8÷8 = 1 whole
A fraction where top equals bottom is always equal to 1, not 0.
Spot the mistakeIs each simplification correct? Click to check.
Check A
16/40 simplified → 4/10
❌ WRONG — not fully simplified. 4/10: HCF(4,10)=2 → divide again → 2/5 is the correct final answer.
A CNC machine completes 45 cuts in 75 seconds. Express the cut-rate as a simplified fraction.
[3]
Rate = 45/75 HCF(45,75)=15 45÷15=3 75÷15=5 Answer: 3/5 cuts per second
Q10
A gear train has a drive gear with 56 teeth and a driven gear with 84 teeth. Express the gear ratio as a simplified fraction and explain what it means.
[4]
Gear ratio = 56/84 HCF(56,84)=28 56÷28=2, 84÷28=3 Answer: 2/3 Meaning: for every 3 turns of the drive gear, the driven gear turns 2 times. The output is slower — geared DOWN.
Q11
An alloy batch uses 180g aluminium and 270g steel. What simplified fraction of the total is aluminium?
[4]
Total = 180+270 = 450g Fraction aluminium = 180/450 HCF(180,450)=90 180÷90=2, 450÷90=5 Answer: 2/5 of the batch is aluminium
Q12
A hydraulic cylinder exerts 1260N over a piston area of 1680mm². Express the pressure as a simplified fraction, then as a decimal.
