# Simplification and Approximation — BODMAS, Squares and Fast Calculation

Source: https://skills101.in/learn/aptitude/simplification-and-approximation — Skills101 free tutorial (Quantitative Aptitude). Updated 2026-11-09.

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

![BODMAS ladder: Brackets, then Orders/Of, then Divide and Multiply left to right, then Add and Subtract left to right](https://skills101.in/learn-img/aptitude/bodmas.svg)

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]*

1. Innermost bracket: 8 − 2 = 6
2. Square bracket: 12 − 6 ÷ 3 = 12 − 2 = 10
3. Multiply: 3 × 10 = 30
4. 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.


Practice quiz and free certificate: https://skills101.in/learn/aptitude/simplification-and-approximation