Mathematical Reasoning · Module 9
Fractions Made Simple
Cross-multiplying, the butterfly method and more — the tricks that make fractions quick, with four tiers of practice.
Page 1 · Introduction
Fractions made simple
Fractions trip up lots of students — but with a few clever tricks, they become quick and even satisfying. This module gives you shortcuts for comparing, simplifying and adding fractions without slow, error-prone working.
The big ideaYou rarely need to find big common denominators. Smart tricks like cross-multiplying and the "butterfly" method do the work in seconds.
💡We build from simplifying up to comparing and adding tricky fractions. Take it a step at a time and the fear disappears.
On the real testWhere this appearsFractions run through the whole Mathematical Reasoning section (25% of your mark, no calculator from 2025). Being quick and confident with them frees up time everywhere.
Page 2 · Simplifying
Making fractions smaller
To simplify, divide the top and bottom by the same number until you can't go further.
Simplify 4/8.
Both divide by 4: 4÷4 = 1, 8÷4 = 2. So 4/8 = ½.
½
Simplify 6/9.
Both divide by 3: 6÷3 = 2, 9÷3 = 3. So 6/9 = ⅔.
⅔
✅Quick check: if top and bottom are both even, you can always divide by 2. Keep going until one of them is odd.
Page 3 · Fraction of a number
Finding a fraction of an amount
To find a fraction of a number: divide by the bottom, multiply by the top.
Find ⅔ of 30.
Divide by 3: 30 ÷ 3 = 10. Multiply by 2: 10 × 2 = 20.
20
Find ¾ of 40.
40 ÷ 4 = 10, × 3 = 30.
30
💡Always divide first, then multiply — the numbers stay smaller and easier to handle that way.
Page 4 · The comparing trick
Which fraction is bigger?
No need for common denominators. Cross-multiply: multiply each top by the other bottom, and compare.
How to do it
To compare ⅔ and ¾: multiply 2 × 4 = 8 (goes with ⅔) and 3 × 3 = 9 (goes with ¾). Since 9 > 8, ¾ is bigger.
✅The bigger cross-product sits above the bigger fraction. Always multiply "upwards and across" so each product stays paired with the right fraction.
Page 5 · The butterfly method
Adding fractions the easy way
The "butterfly" method adds two fractions without hunting for a common denominator.
The three steps
For ½ + ⅓:
- New bottom: multiply the bottoms: 2 × 3 = 6.
- New top: cross-multiply and add: (1×3) + (1×2) = 3 + 2 = 5.
- Answer: 5/6.
¼ + ⅔
Bottom: 4×3 = 12. Top: (1×3)+(2×4) = 3+8 = 11. Answer 11/12.
11/12
🌟It's called the butterfly because the cross-multiply lines look like wings. It works for subtracting too — just subtract the tops instead of adding.
Page 6 · Multiplying fractions
The easiest operation of all
Multiplying fractions is actually the simplest thing you can do — just multiply straight across.
top × top, bottom × bottom
⅔ × ¾
Tops: 2×3 = 6. Bottoms: 3×4 = 12. So 6/12 = ½.
½
TipSimplify at the end — 6/12 becomes ½. Multiplying is the one time you don't need matching bottoms.
Page 7 · Practice
Level 1 · Easy
Warm up
Simplifying and fractions of amounts. Tap to check.
1. Simplify 4/8.
Divide both by 4: ½.
2. Simplify 6/9.
Divide both by 3: ⅔.
3. What is ½ of 20?
Half of 20 = 10.
🎉Simplifying is just dividing top and bottom by the same number. Now let's compare fractions.
Page 8 · Practice
Level 2 · Medium
Comparing & finding
Use cross-multiplying to compare, and divide-then-multiply to find.
1. Which is bigger: ⅔ or ¾?
A ⅔
B ¾
C Equal
D Can't tell
Cross-multiply: 2×4 = 8 (for ⅔), 3×3 = 9 (for ¾). 9 > 8, so ¾ is bigger.
2. Which is bigger: ⅗ or 5/8?
A ⅗
B Equal
C 5/8
D Can't tell
Cross-multiply: 3×8 = 24 (for ⅗), 5×5 = 25 (for 5/8). 25 > 24, so 5/8 is bigger.
3. What is ⅖ of 30?
30 ÷ 5 = 6, × 2 = 12.
💪Cross-multiplying beats finding common denominators every time for a quick comparison. A real time-saver.
Page 9 · Practice
Level 3 · Hard
The butterfly method
Add and subtract with the butterfly — new bottom = bottoms multiplied, new top = cross-multiply and add (or subtract).
1. ½ + ⅓ = ?
Bottom 2×3 = 6. Top (1×3)+(1×2) = 5. Answer 5/6.
2. ¼ + ⅔ = ?
Bottom 4×3 = 12. Top (1×3)+(2×4) = 3+8 = 11. Answer 11/12.
3. ¾ − ½ = ?
Bottom 4×2 = 8. Top (3×2)−(1×4) = 6−4 = 2. So 2/8 = ¼.
💡The butterfly works for any two fractions, no matter how awkward the denominators. Master it and adding fractions never scares you again.
Page 10 · Challenge
Level 4 · Challenge
Think smart
These combine tricks. Choose the fastest tool for each.
1. Which is bigger: 5/7 or 7/9?
A 5/7
B Equal
C 7/9
D Can't tell
Cross-multiply: 5×9 = 45 (for 5/7), 7×7 = 49 (for 7/9). 49 > 45, so 7/9 is bigger.
2. What is ⅔ of ¾?
"Of" means multiply: (2×3)/(3×4) = 6/12 = ½.
3. ½ + ¼ + ⅛ = ?
Use eighths: ½ = 4/8, ¼ = 2/8, ⅛ = 1/8. Add: 4+2+1 = 7/8. Spotting that all convert neatly to eighths beats the butterfly here.
🌟Notice the last one was faster with a common denominator than the butterfly — choosing the smartest method for each question is the real skill.
Complete
Superb!
You've completed the Fractions module.
You can now simplify quickly, find fractions of amounts, compare with cross-multiplying, add and subtract with the butterfly method, and multiply straight across. Fractions just became one of your strengths.