Simplification and Approximation Notes

Simplification and Approximation is the speed-building chapter of Quantitative Aptitude. In SSC CGL, CHSL, bank prelims and state exams, 5 to 15 questions can come only from this topic, and the same calculation skills help in every other chapter, from percentages to data interpretation. In UPSC CSAT, direct simplification questions are fewer, but fast and accurate calculation saves precious minutes in Paper II. The good news is that this topic needs no deep theory: it needs a fixed order of operations, a few memorised values and regular practice.

Quick Facts: Simplification at a Glance

• Order of operations: BODMAS or VBODMAS (Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction).
• Must memorise: squares up to 30, cubes up to 20, fraction-decimal values up to 1/20, and basic surd values.
• Approximation: Round numbers to the nearest convenient value, then compare with options.
• Checking tool: Unit digit and digit-sum (casting out nines) methods help eliminate wrong options.

1. The BODMAS / VBODMAS Rule

When an expression has many operations, you must follow a fixed order. Otherwise, the same question gives different answers.

Important: Division and multiplication have equal priority in practice; when both appear, solve from left to right. The same applies to addition and subtraction.

Example 1: 48 ÷ 12 × 3 + 6 − 2 = 4 × 3 + 6 − 2 = 12 + 6 − 2 = 16.

Example 2: 1/2 of 60 ÷ 5 = 30 ÷ 5 = 6 (because "of" is solved before division).

Example 3: 18 − [5 − {6 + 2 (7 − 8 − 5)}]. Step by step: 7 − 8 − 5 = −6 (bar), 2 × (−6) = −12, 6 − 12 = −6, 5 − (−6) = 11, 18 − 11 = 7.

2. Fractions and Decimals You Must Know

Converting fractions to decimals (and percentages) instantly is the biggest time saver.

Recurring decimals: 0.777... = 7/9; 0.2727... = 27/99 = 3/11; 0.1666... = (16 − 1)/90 = 1/6. Rule: write the repeating part over as many 9s as there are repeating digits; for mixed recurring decimals, subtract the non-repeating part and use 9s followed by 0s.

3. Squares, Cubes and Roots

• Squares up to 30: e.g. 13² = 169, 17² = 289, 23² = 529, 27² = 729.
• Cubes up to 20: e.g. 12³ = 1728, 15³ = 3375, 18³ = 5832.
• Square of numbers ending in 5: 65² → 6 × 7 = 42, attach 25 → 4225.
• Numbers near 100: 97² → 97 − 3 = 94, 3² = 09 → 9409. 104² → 104 + 4 = 108, 4² = 16 → 10816.
• Square roots by unit digit: A perfect square ending in 6 has a root ending in 4 or 6. Use the nearby squares to choose.

Example: √7056. Unit digit 6 → root ends in 4 or 6. 80² = 6400 and 90² = 8100, so the root is in the 80s. 85² = 7225 is greater, so the answer is 84.

4. Surds and Indices

Surds are roots that cannot be simplified into whole numbers, like √2 or √3. Basic rules:

• a^m × aⁿ = a^m+n; a^m ÷ aⁿ = a^m−n; (a^m)ⁿ = a^mn.
• a⁰ = 1 and a⁻ⁿ = 1/aⁿ.
• √a × √b = √(ab), but √a + √b is not √(a + b).
• Useful values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236.
• Rationalisation: 1/(√3 − √2) = (√3 + √2)/(3 − 2) = √3 + √2.

5. Useful Algebraic Identities for Calculation

• (a + b)² = a² + 2ab + b² → 102² = 10000 + 400 + 4 = 10404.
• a² − b² = (a + b)(a − b) → 53² − 47² = 100 × 6 = 600.
• (a + b)(a − b) → 48 × 52 = 50² − 2² = 2496.
• a³ + b³ = (a + b)(a² − ab + b²), useful in fraction simplification.

6. Approximation: The Smart Shortcut

In approximation questions, options are usually far apart, so exact calculation is not needed. Round the numbers to friendly values and calculate quickly.

• Round decimals to the nearest whole number: 39.98 → 40, 15.02 → 15.
• Round percentages: 24.95% of 801 → 25% of 800 = 200.
• Round square roots to nearest perfect square: √626 → √625 = 25.
• Round both numerator and denominator in the same direction where possible to reduce error.

Example: 49.98% of 1199.9 + 15.02 × 6.97 ≈ 50% of 1200 + 15 × 7 = 600 + 105 = 705.

Example: (3999 ÷ 79.9) × 4.02 ≈ (4000 ÷ 80) × 4 = 50 × 4 = 200.

7. Checking Answers: Unit Digit and Digit Sum

• Unit digit method: In 237 × 418, the answer must end in 7 × 8 = 56 → 6. Remove options not ending in 6.
• Digit sum method: Reduce every number to a single digit by adding its digits (e.g., 457 → 16 → 7). The digit sum of the answer must match the digit sum of the operation result. This catches most calculation errors.

Common Mistakes to Avoid

1. Doing addition before division just because it appears first.
2. Ignoring the vinculum (bar) or solving brackets in the wrong order.
3. Over-rounding in approximation, where options are close together.
4. Writing √a + √b = √(a + b).
5. Not converting mixed fractions like 2 1/3 into improper fractions (7/3) before multiplying.

Study Plan for This Topic

• Week 1: Memorise squares, cubes and the fraction table. Revise for 10 minutes daily.
• Week 2: Solve 20 BODMAS and 20 approximation questions a day with a timer.
• Week 3 onwards: Take sectional mock tests. Aim to solve each simplification question in 30 to 40 seconds.

Good calculation is a habit, not a talent. Ten focused minutes every day will make this chapter one of your surest sources of marks in any competitive exam.

Solved Practice Set

Work through these questions in order. Each one tests a different idea from this chapter, so read the method even if your answer is correct.

1. Simplify: 120 ÷ 8 of 3 + 25 × 2.
   Solution: "Of" first: 8 of 3 = 24. Then 120 ÷ 24 = 5. Next 25 × 2 = 50. Answer = 5 + 50 = 55.
2. Simplify: 3/4 + 5/6 − 7/12.
   Solution: LCM of 4, 6 and 12 is 12. So 9/12 + 10/12 − 7/12 = 12/12 = 1.
3. Find: 0.36 written as a fraction (6 recurring).
   Solution: (36 − 3) ÷ 90 = 33/90 = 11/30.
4. Find: 98 × 102.
   Solution: (100 − 2)(100 + 2) = 10000 − 4 = 9996.
5. Approximate: 34.97% of 599.8 + √(1023) ≈ ?
   Solution: 35% of 600 = 210; √1024 = 32. Answer ≈ 242.
6. Approximate: 2999 ÷ 24.98 × 5.01 ≈ ?
   Solution: 3000 ÷ 25 = 120; 120 × 5 = 600.
7. Simplify: (√12 + √27) ÷ √3.
   Solution: √12 = 2√3 and √27 = 3√3. Sum = 5√3. Divided by √3 = 5.
8. Find the value: 2⁵ × 2⁻³ ÷ 2⁰.
   Solution: 2^{5 − 3} = 2² = 4, and 2⁰ = 1. Answer = 4.

How Simplification Helps in Other Chapters

Strong simplification skills support almost every chapter of aptitude. In percentage questions, knowing that 12.5% = 1/8 turns a long multiplication into one division. In data interpretation, approximation helps you compare growth rates across years without exact calculation. In profit and loss and interest questions, fraction values make successive changes easy to handle. In geometry and mensuration, surd values such as √3 ≈ 1.732 are needed for triangle and hexagon areas. This is why toppers treat simplification as a daily warm-up rather than a one-time chapter.

Related Notes

• Number System Notes: Divisibility, HCF, LCM and Remainders
• Percentage Notes: Formulas, Tricks and Question Types
• Ratio and Proportion Notes
• Percentage, Ratio and Average for CSAT

Frequently Asked Questions

What is the BODMAS rule?

BODMAS fixes the order of operations: Brackets, Of, Division, Multiplication, Addition and Subtraction. Division and multiplication, and addition and subtraction, are solved from left to right.

What is VBODMAS?

VBODMAS adds Vinculum (a bar over numbers) at the start. The expression under the bar is solved before brackets.

What does "of" mean in simplification?

"Of" means multiplication, but it is solved before division and ordinary multiplication.

What is the difference between simplification and approximation?

Simplification asks for an exact value. Approximation asks for a near value, so you can round numbers to make calculation faster.

How many squares and cubes should I memorise?

Memorise squares up to 30 and cubes up to 20. Many students also learn squares up to 50 for faster calculation.

How do you convert a recurring decimal into a fraction?

Write the repeating digits over the same number of 9s. For example, 0.454545... = 45/99 = 5/11.

What is the quickest way to square a number ending in 5?

Multiply the tens part by the next number and attach 25. For 75², 7 × 8 = 56, so the answer is 5625.

How can I check my answer quickly?

Use the unit digit method or the digit-sum method to remove wrong options without full calculation.

Is simplification asked in UPSC CSAT?

Direct simplification questions are few in CSAT, but fast calculation is essential for percentages, data interpretation and number system questions.

How many simplification questions come in SSC and bank exams?

It varies by exam and year, but bank prelims often have around 5 to 15 questions and SSC exams also include several calculation-based questions.

What are the values of √2 and √3?

√2 is about 1.414 and √3 is about 1.732. These values are useful in surds and geometry questions.

How can I improve my calculation speed?

Memorise key values, practise daily with a timer, use identities like a² − b², and avoid writing unnecessary steps.