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Simplification & Approximation

The 90-Second Trap That Decides Your Rank

In IBPS PO Mains, a candidate once spent 4 minutes solving a single simplification question — and got it wrong. Another candidate glanced at the same question, rounded the numbers, and picked the answer in 20 seconds. The difference wasn't intelligence. It was strategy.

Simplification and Approximation questions look like pure arithmetic, but they are actually tests of decision-making under time pressure. Exams like SSC CGL Tier-II, IBPS PO, SBI PO, and CSAT use them to filter out candidates who calculate everything from those who calculate *smartly*.

Learning goals for this chapter:

• Master BODMAS and its hidden traps
• Learn when to approximate and when *not* to
• Build a toolkit of fast-calculation techniques (digit sums, fractions, percentages)
• Recognise the examiner's favourite traps before they catch you

1. The Foundation: BODMAS Done Right

Every simplification question begins with the order of operations:

Brackets → Orders (powers, roots) → Division → Multiplication → Addition → Subtraction

But here's what most students miss: Division and Multiplication have equal priority — solve them left to right. Same for Addition and Subtraction.

Example: 24 ÷ 4 × 2 = ?

• Wrong: 24 ÷ (4 × 2) = 3
• Right: (24 ÷ 4) × 2 = 6 × 2 = 12

Also remember the bracket hierarchy: Vinculum (bar) → ( ) → { } → [ ]. The bar is the smallest and is solved first.

🧠 Active Recall: Without looking back — which two operations share equal priority, and in what order do you solve them? What comes before round brackets?

2. The Golden Question: Exact or Approximate?

Before touching a single number, look at the answer options. This one habit separates toppers from the rest.

Approximate when:

• Options are far apart (e.g., 120, 480, 950, 1800)
• The question says "approximately" or "≈"
• Numbers are ugly decimals like 19.97, 49.8, 24.9%

Calculate exactly when:

• Options are close (e.g., 246, 248, 250, 252)
• The question involves exact fractions or surds that cancel

Rule of thumb: If the gap between options is more than 5% of the values, approximation is safe. If options differ by 1–2%, calculate precisely.

🧠 Active Recall: You see options 118, 240, 365, 490. Should you approximate? What if the options were 240, 242, 244, 246?

3. The Approximation Toolkit

Technique 1: Round to Friendly Numbers

19.97 × 5.03 ≈ 20 × 5 = 100
498 ÷ 24.9 ≈ 500 ÷ 25 = 20

Technique 2: Percentage ↔ Fraction Conversions

Memorise these — they appear constantly:

Example: 37.5% of 640 = (3/8) × 640 = 240 — done in 3 seconds.

Technique 3: Break Multiplication Apart

17 × 23 = 17 × 20 + 17 × 3 = 340 + 51 = 391
48 × 25 = 48 × 100/4 = 4800/4 = 1200

🧠 Active Recall: What is 62.5% as a fraction? How would you compute 36 × 25 in under 5 seconds?

4. The Digit Sum Check (Casting Out Nines)

This is your verification weapon. Add the digits of a number repeatedly until one digit remains. The digit sum of a correct answer must match the digit sum of your computed expression.

Example: Is 347 × 26 = 9,022 correct?

• Digit sum of 347: 3+4+7 = 14 → 5
• Digit sum of 26: 8
• 5 × 8 = 40 → 4
• Digit sum of 9,022: 9+0+2+2 = 13 → 4 ✓ (consistent)

Caution: Digit sums can confirm but not guarantee — a wrong answer can occasionally match. Use it to *eliminate* options, not to prove.

Mnemonic — "NINE is FINE": When casting out nines, you can drop any 9s or digit pairs summing to 9 instantly. In 9,022, the 9 vanishes immediately: 0+2+2 = 4.

🧠 Active Recall: Use digit sums to check: does 128 × 13 = 1,664 pass the test?

5. Surds, Squares and Square Roots

Advanced papers love mixing surds into simplification.

Key facts to have at your fingertips:

• Squares of 1–30 (at minimum 1–25): 17² = 289, 23² = 529, 29² = 841
• √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236
• (a + b)² = a² + 2ab + b² and (a − b)(a + b) = a² − b²

Fast square trick (numbers ending in 5): 65² = 6 × 7 | 25 = 4225. Multiply the tens digit by (itself + 1), then append 25.

Near-100 squares: 98² = (100 − 2)² = 10000 − 400 + 4 = 9604

🧠 Active Recall: Compute 75² and 103² using the tricks above. What is √3 approximately?

6. The Examiner's Favourite Traps

Trap 1 — The left-to-right illusion: 100 − 50 + 25. Students who group (50 + 25) get 25; the correct left-to-right answer is 75.

Trap 2 — Over-approximation: Rounding 49.8% of 199.6 to 50% of 200 = 100 is fine when options are 60, 100, 140, 180 — but dangerous when options are 98, 100, 102, 104.

Trap 3 — Decimal drift: 0.1 + 0.2 × 0.3. Multiplication first: 0.1 + 0.06 = 0.16, not 0.09.

Trap 4 — Negative sign slips: In expressions like 45 − (12 − 18), the inner bracket gives −6, so the answer is 45 + 6 = 51.

🧠 Active Recall: Solve 80 − 30 + 20 left to right. Then solve 0.4 + 0.6 × 0.5. Which trap does each illustrate?

7. A Worked Advanced Example

Question: 24.96% of 799.8 + 15.9 × 4.1 − 49.7 ≈ ?

Step 1 — Scan options first. Assume they're widely spaced, so approximate.

Step 2 — Round: 25% of 800 + 16 × 4 − 50

Step 3 — Compute: (1/4) × 800 = 200; 16 × 4 = 64; so 200 + 64 − 50 = 214

Total time: under 30 seconds. The exact calculation would take 3 minutes and risk arithmetic slips.

Summary Table

🔁 Review This After 2 Days

Close the book and answer from memory:

1. What is the full BODMAS order, and which pairs share priority?
2. When is approximation unsafe?
3. Write the fractions for 12.5%, 37.5%, 62.5%, 83.33%.
4. Compute 95² and 62.5% of 480 mentally.
5. What does the digit-sum check prove — and what can it *not* prove?

If you stumble on any, re-read that section today — not tomorrow.