Quantitative Aptitude · Lesson 3 of 18 · Updated ·
Simplification and Approximation — BODMAS, Squares and Fast Calculation
Simplification questions look scary — long lines of numbers and brackets — but they test only two things: doing operations in the right order and calculating fast. Speed here gives you extra minutes for harder questions.
1. BODMAS: the order of operations
The big trap: Division and Multiplication have the same rank, so you go left to right. The same for Addition and Subtraction.
- 24 ÷ 4 × 3 = 6 × 3 = 18 ✓ (not 24 ÷ 12 = 2 ✗)
- 20 − 5 + 3 = 15 + 3 = 18 ✓ (not 20 − 8 = 12 ✗)
"Of" means multiply, and it comes before ÷. ½ of 40 ÷ 4 = 20 ÷ 4 = 5.
Brackets go from inside out: ( ) first, then { }, then [ ].
Example 1
Simplify: 18 + 3 × [12 − (8 − 2) ÷ 3]
- Innermost bracket: 8 − 2 = 6
- Square bracket: 12 − 6 ÷ 3 = 12 − 2 = 10
- Multiply: 3 × 10 = 30
- Add: 18 + 30 = 48
2. Learn these by heart
Squares 11–30: 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900.
Cubes 1–15: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375.
Square roots: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236.
3. Multiplication shortcuts
| Trick | How | Example |
|---|---|---|
| × 5 | × 10, then ÷ 2 | 68 × 5 = 680 ÷ 2 = 340 |
| × 25 | × 100, then ÷ 4 | 48 × 25 = 4800 ÷ 4 = 1200 |
| × 11 (2-digit) | Write the two digits, put their sum in the middle | 53 × 11 = 5 (5+3) 3 = 583 |
| × 99 | × 100, then subtract the number | 47 × 99 = 4700 − 47 = 4653 |
| Square ending in 5 | n(n+1) then write 25 | 65² → 6 × 7 = 42 → 4225 |
| Near 100 | (100 + a)(100 + b) = 100(100 + a + b) + ab | 103 × 106 = 100 × 109 + 18 = 10918 |
Two algebra identities save a lot of time:
- (a + b)² = a² + 2ab + b² → 102² = 10000 + 400 + 4 = 10404
- (a + b)(a − b) = a² − b² → 49 × 51 = 50² − 1² = 2499
Example 2
Find 998 × 1002.
= (1000 − 2)(1000 + 2) = 1,000,000 − 4 = 999,996.
4. Approximation
In "approximately equal to" questions the options are usually far apart, so round each number to something easy and calculate.
Example 3
What is approximately 49.87% of 601.2 + 19.98 × 15.03?
≈ 50% of 600 + 20 × 15 = 300 + 300 = 600.
Rules for safe rounding:
- Round to numbers that make the maths easy (50%, 600, 20).
- If you round one number up, try to round another down, so errors cancel.
- If two options are close, round less aggressively and re-check.
5. Fractions and decimals quickly
- To compare fractions, cross-multiply: is 5/8 > 7/11? 5 × 11 = 55, 7 × 8 = 56, so 5/8 < 7/11.
- Recurring decimals: 0.3̄ = 3/9 = 1/3, 0.4̄5̄ = 45/99 = 5/11.
Common mistakes
- Doing × before ÷ just because "M" comes after "D" in BODMAS. They are equal — go left to right.
- Forgetting that "of" is done before division.
- Over-rounding when two options are close.
Quick recap
- BODMAS: Brackets → Orders/Of → ÷ × (left to right) → + − (left to right).
- Memorise squares to 30 and cubes to 15.
- Use (a+b)(a−b) = a² − b² and the near-100 trick for big products.
- Approximate by rounding to easy numbers; balance ups and downs.