In this guide
Simplification is arithmetic done in the right order and done fast. The questions need no formula and no clever idea, which is exactly why they punish carelessness. Examiners know the common slips, and the answer you get from each slip is usually sitting among the four options.
It is also the quickest area of Maths to improve. A week of focused practice on order of operations, fraction–decimal pairs and roots can make these questions take 20 to 30 seconds each. And the same skills run through every other topic: a profit and loss or interest question ends in a simplification.
BODMAS, and why the order is fixed
| Step | Stands for | Note |
|---|---|---|
| B | Brackets | Innermost first: ( ), then { }, then [ ] |
| O | Orders and "of" | Powers, roots, and "of" |
| D, M | Division and Multiplication | Equal rank, left to right |
| A, S | Addition and Subtraction | Equal rank, left to right |
Why left to right matters: division is multiplication by a reciprocal, so 18 ÷ 3 × 2 means 18 × 1/3 × 2 = 12. If you did the multiplication first you would get 18 ÷ 6 = 3, a different number. The rule exists so that one expression has one answer.
The "of" rule. In Indian exam papers, "of" is treated as multiplication that comes before ordinary division and multiplication. So 48 ÷ 4 of 3 = 48 ÷ 12 = 4, not 12 × 3 = 36. "Of" binds the two numbers beside it into one quantity, as in "one-third of 90".
Fractions and decimals: swap to whichever is faster
Many decimals are friendly fractions in disguise. Learn these pairs and multiplication becomes division by a small number.
| Decimal | Fraction | Decimal | Fraction |
|---|---|---|---|
| 0.5 | 1/2 | 0.125 | 1/8 |
| 0.25 | 1/4 | 0.375 | 3/8 |
| 0.75 | 3/4 | 0.625 | 5/8 |
| 0.2 | 1/5 | 0.0625 | 1/16 |
| 0.333… | 1/3 | 0.1666… | 1/6 |
So 0.375 × 640 is 3/8 of 640 = 3 × 80 = 240, with no decimal work at all.
Approximation: when to round and when not to
Some questions say "approximately" and give options far apart. Round each number to a convenient value, then calculate.
- 399.8 × 20.1 ≈ 400 × 20 = 8,000.
- 1,198 ÷ 39.9 ≈ 1,200 ÷ 40 = 30.
Why it is safe: rounding 399.8 up by 0.2 and 20.1 down by 0.1 changes the product by well under 1%. If the options differ by 10% or more, that error cannot change your choice.
Look at the options first. If two options are close, round less, or calculate exactly.
Squares, square roots and cube roots
Know squares to 30 and cubes to 15 by heart. Then use the unit digit to find roots of perfect squares and cubes.
Square roots. A square ending in 1 has a root ending in 1 or 9; ending in 4, root ends in 2 or 8; ending in 5, root ends in 5; ending in 6, root ends in 4 or 6; ending in 9, root ends in 3 or 7.
For √1,764: 40² = 1,600 and 50² = 2,500, so the root is in the 40s and ends in 2 or 8. 1,764 is much nearer 1,600, so try 42: 42² = 1,764.
Cube roots. Each unit digit of a cube points to exactly one unit digit of the root. 0, 1, 4, 5, 6 and 9 stay the same; 2 and 8 swap; 3 and 7 swap.
For ∛2,197: ignore the last three digits and look at 2. The largest cube not above 2 is 1³, so the tens digit is 1. The unit digit 7 comes from a root ending in 3. Answer: 13. Check: 13³ = 2,197.
Identities that save time
- (a + b)² = a² + 2ab + b², so 102² = 10,000 + 400 + 4 = 10,404.
- (a − b)² = a² − 2ab + b², so 98² = 10,000 − 400 + 4 = 9,604.
- a² − b² = (a + b)(a − b), so 53² − 47² = 100 × 6 = 600.
- Near 100: 97 × 96 = (100 − 3 − 4) hundreds + 3 × 4 = 9,312.
Why the last one works: (100 − a)(100 − b) = 100(100 − a − b) + ab.
Worked questions at NTPC level
Q1. Simplify 18 ÷ 3 × 2 + 4 − 1.
Left to right for ÷ and ×: 18 ÷ 3 = 6, then 6 × 2 = 12. Then 12 + 4 − 1 = 15.
Q2. Simplify 96 ÷ 4 of 2 + 3 × 5.
"Of" first: 4 of 2 = 8. Then 96 ÷ 8 = 12 and 3 × 5 = 15. Total: 12 + 15 = 27.
Q3. Find the value of 2/5 of 3/4 of 1,200 + 15% of 400.
3/4 of 1,200 = 900. 2/5 of 900 = 360. 15% of 400 = 60. Total: 420.
Q4. Find the value of 0.25 × 480 + 0.125 × 640.
0.25 = 1/4 and 0.125 = 1/8. So 480 ÷ 4 + 640 ÷ 8 = 120 + 80 = 200.
Q5. Find the approximate value of 49.8% of 601 + 19.95 × 15.02.
Round: 50% of 600 = 300, and 20 × 15 = 300. Answer ≈ 600.
(The exact value is about 598.9, so 600 is the right option.)
Q6. Find √1,764 + ∛2,197.
From the sections above: 42 + 13 = 55.
Common mistakes
- Multiplying before a division that comes first on the left. 48 ÷ 6 × 3 is 24, not 48 ÷ 18.
- Ignoring "of". 72 ÷ 3 of 4 is 72 ÷ 12 = 6.
- Rounding when options are close. Approximate only when the options are clearly apart.
- Dropping a minus sign when you open a bracket: 20 − (8 − 3) = 15, not 9.
- Copying a digit wrongly from the screen to your rough sheet. Read the number twice.
Practice set
- 48 ÷ 6 × 3 − 4
- 7 + 2 × (15 − 5) ÷ 4
- √2,025
- 98²
- 40% of 250 + 3/4 of 120
- 65² − 35²
- 72 ÷ 3 of 4 + 10
- Approximately: 29.9 × 40.2 − 199.8
Answers:
- 20. 48 ÷ 6 = 8; 8 × 3 = 24; 24 − 4 = 20.
- 12. Bracket 10; 2 × 10 = 20; 20 ÷ 4 = 5; 7 + 5 = 12.
- 45. Between 40² and 50², ends in 5; 45² = 2,025.
- 9,604. (100 − 2)² = 10,000 − 400 + 4.
- 190. 100 + 90.
- 3,000. (65 + 35)(65 − 35) = 100 × 30.
- 16. 3 of 4 = 12; 72 ÷ 12 = 6; 6 + 10 = 16.
- About 1,000. 30 × 40 − 200 = 1,000. (Exact: about 1,002.)
What to do next
- Learn the decimal–fraction table above until you can say each pair without thinking.
- Solve 20 simplification questions a day for a week, with a 30-second target for each.
- Practise ten square roots and five cube roots daily using the unit-digit method.
- Move on to percentage, which uses the same fraction pairs, and keep the topic in your maths plan.
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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