In this guide
Simplification is about doing arithmetic in the right order. The numbers are small and the methods are easy, which is exactly why these questions catch people out: a candidate rushes, does one step in the wrong order, gets a clean-looking answer, and finds it waiting among the options. Wrong options are often exactly what the common wrong orders produce.
It is also one of the quickest areas to improve. Learn the order below, practise twenty questions, and the marks are yours.
The BODMAS order
| Order | Letter | Stands for |
|---|---|---|
| 1 | B | Brackets |
| 2 | O | Of, and orders (powers and roots) |
| 3 | D M | Division and Multiplication, from left to right |
| 4 | A S | Addition and Subtraction, from left to right |
Brackets inside brackets
Work from the innermost bracket outwards. The usual order is ( ) first, then { }, then [ ]. A line drawn over a group of numbers (a "bar") works like a bracket and is done before everything else.
The "of" rule
"Of" means multiply, but in BODMAS it is done before ordinary division and multiplication. So "48 ÷ 1/2 of 8" and "48 ÷ 1/2 × 8" give different answers. Worked example 3 shows both.
Worked examples
Example 1: 24 ÷ 4 × 3 − 5
- Division and multiplication, left to right: 24 ÷ 4 = 6, then 6 × 3 = 18.
- Then subtraction: 18 − 5 = 13.
- The trap: doing 4 × 3 first gives 24 ÷ 12 = 2, and 2 − 5 = −3. Wrong.
Example 2: 10 + 2 × (9 − 3) ÷ 4
- Bracket: 9 − 3 = 6.
- Left to right: 2 × 6 = 12, then 12 ÷ 4 = 3.
- 10 + 3 = 13.
Example 3 (the "of" trap): Find 48 ÷ 1/2 of 8, and then 48 ÷ 1/2 × 8.
- With "of": 1/2 of 8 = 4 first. Then 48 ÷ 4 = 12.
- With "×": left to right. 48 ÷ 1/2 = 96, then 96 × 8 = 768.
- One small word changes the answer from 12 to 768.
Example 4 (nested brackets): 50 − [20 − {8 + (6 − 2) × 2}]
- Innermost: 6 − 2 = 4.
- Inside { }: 8 + 4 × 2. Multiply first: 4 × 2 = 8, then 8 + 8 = 16.
- Inside [ ]: 20 − 16 = 4.
- Finally: 50 − 4 = 46.
Example 5 (powers and roots): √144 + 3² × 2 − 10 ÷ 5
- Roots and powers first: √144 = 12, 3² = 9.
- Now 12 + 9 × 2 − 10 ÷ 5.
- Multiply and divide: 9 × 2 = 18, 10 ÷ 5 = 2.
- 12 + 18 − 2 = 28.
Example 6 (percentages and fractions): 3/4 of 240 − 20% of 150 + 125% of 80
- 3/4 of 240 = 180.
- 20% of 150 = 30.
- 125% of 80 = 5/4 × 80 = 100.
- 180 − 30 + 100 = 250.
Percentages as fractions
Many simplification questions mix percentages and fractions. These pairs turn them into one-step sums:
| Percentage | Fraction | Percentage | Fraction |
|---|---|---|---|
| 12.5% | 1/8 | 50% | 1/2 |
| 20% | 1/5 | 62.5% | 5/8 |
| 25% | 1/4 | 75% | 3/4 |
| 33⅓% | 1/3 | 125% | 5/4 |
| 37.5% | 3/8 | 150% | 3/2 |
Quick multiplication
Use (a + b)(a − b) = a² − b²:
- 97 × 103 = (100 − 3)(100 + 3) = 10,000 − 9 = 9,991.
- 48 × 52 = (50 − 2)(50 + 2) = 2,500 − 4 = 2,496.
Use (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²:
- 101² = 10,000 + 200 + 1 = 10,201.
- 99² = 10,000 − 200 + 1 = 9,801.
Multiplying by 5 or 25:
- × 5 is the same as × 10 ÷ 2: 468 × 5 = 4,680 ÷ 2 = 2,340.
- × 25 is the same as × 100 ÷ 4: 64 × 25 = 6,400 ÷ 4 = 1,600.
Approximation
When the options are far apart, round the numbers and pick the closest option:
- 49.8 × 30.2 ≈ 50 × 30 = 1,500. (The exact answer is 1,503.96.)
- 1,203 ÷ 39.8 ≈ 1,200 ÷ 40 = 30.
If the options are close together, for example 1,498, 1,504 and 1,510, don't round. Calculate exactly.
Common mistakes
- Doing multiplication before a division that comes first on the left.
- Treating "of" like ordinary ×, or ordinary × like "of".
- Starting with the outer bracket instead of the innermost one.
- Adding before a subtraction that comes first on the left: 20 − 5 + 3 is 18, not 12.
- Rounding when the options are close.
Practice set
- 36 ÷ 4 × 2 + 7
- 15 − 3 × (8 − 5) + 4
- 2/5 of 350 + 10% of 90
- 98 × 102
- 102²
- √1,296
- 64 ÷ 1/4 of 16
- 80 − [30 − {12 − (9 − 5)}]
Answers:
- 25. 36 ÷ 4 = 9; 9 × 2 = 18; 18 + 7 = 25.
- 10. Bracket 8 − 5 = 3; 3 × 3 = 9; 15 − 9 + 4 = 10.
- 149. 2/5 of 350 = 140; 10% of 90 = 9; 140 + 9 = 149.
- 9,996. (100 − 2)(100 + 2) = 10,000 − 4.
- 10,404. 10,000 + 400 + 4.
- 36. 36 × 36 = 1,296.
- 16. "Of" first: 1/4 of 16 = 4; then 64 ÷ 4 = 16.
- 58. 9 − 5 = 4; 12 − 4 = 8; 30 − 8 = 22; 80 − 22 = 58.
What to do next
- Write the BODMAS table and the "of" rule on the first page of your maths notebook.
- Learn the percentage–fraction pairs above until you can say them without looking.
- Solve 20 simplification questions from past papers, timing each set of five.
- Go back to the number system for divisibility and remainders, and see the maths plan for what comes next.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Railway Recruitment Boards website .
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