Simplification: BODMAS, Brackets, Roots and Continued Fractions
Simplification: BODMAS, Brackets, Roots and Continued Fractions
Simplification questions reward a fixed order of operations and a few number facts learnt by heart: squares to 30, cubes to 20 and common decimal roots. NTPC sets fraction chains with "of" and brackets, expressions with square, cube and fourth roots, decimal products, and continued fractions.
1. Order of Operations (BODMAS / VBODMAS)
| Order | Operation |
|---|---|
| 1 | Vinculum (bar) and Brackets: ( ), then { }, then [ ] |
| 2 | "Of" (multiplication written as "of") |
| 3 | Division and Multiplication, left to right |
| 4 | Addition and Subtraction, left to right |
"Of" binds tighter than ÷: 12 ÷ 3 of 2 = 12 ÷ 6 = 2, but 12 ÷ 3 × 2 = 8
Division and multiplication have equal rank: work left to right
2. Number Facts to Know Cold
| n | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 21 | 24 | 25 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 121 | 144 | 169 | 196 | 225 | 256 | 289 | 324 | 361 | 441 | 576 | 625 |
| n | 6 | 7 | 8 | 9 | 11 | 12 | 13 | 14 | 15 | 16 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| n³ | 216 | 343 | 512 | 729 | 1331 | 1728 | 2197 | 2744 | 3375 | 4096 | 5832 |
Fourth powers: 2⁴ = 16, 3⁴ = 81, 4⁴ = 256, 5⁴ = 625, 6⁴ = 1296
Decimal roots: √0.09 = 0.3, √0.0016 = 0.04, √0.0049 = 0.07, ∛0.008 = 0.2, ∛0.027 = 0.3
Decimal products: count the total decimal places; 0.08 × 0.6 = 0.048
✗ 18 ÷ 3 × 2 = 18 ÷ 6 = 3 | ✓ Equal rank, left to right: 6 × 2 = 12
✗ √0.0049 = 0.007 | ✓ Pair the decimal places: √0.0049 = 0.07
हिंदी नोट: सरलीकरण में क्रम: कोष्ठक, "का" (of), भाग और गुणा बाएँ से दाएँ, फिर जोड़ और घटाव। "का" भाग से पहले हल होता है। 30 तक के वर्ग और 20 तक के घन याद रखिए।
Exam Pointer: Verified NTPC patterns: fraction chains with ÷, × and brackets (January 2021 CBT-1) and expressions with cube and fourth roots (June 2025 Graduate CBT-1). Nested brackets, decimal products and continued fractions had no verified NTPC shift in our check and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: BODMAS with fractions and "of"
[PYQ: NTPC CBT-1 7-Jan-2021 Shift-1]
EXAM LEVEL
Q. Simplify 18 ÷ 3 × 2 + 4 − 6 ÷ 2.
Left to right for ÷ and ×: 18 ÷ 3 = 6, then 6 × 2 = 12. 6 ÷ 2 = 3. So 12 + 4 − 3 = 13.
Answer: 13
EXAMATLAS LEVEL
Q. Simplify 3/4 of 16 ÷ [2/3 × (1/2 + 1/4)] − 5.
The bracket first: 1/2 + 1/4 = 3/4, and 2/3 × 3/4 = 1/2. "Of" next: 3/4 of 16 = 12. Then 12 ÷ 1/2 = 24, and 24 − 5 = 19.
Answer: 19
Pattern 2: Nested brackets
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Simplify 20 − [12 − {8 − (6 − 4)}].
Innermost first: 6 − 4 = 2, then 8 − 2 = 6, then 12 − 6 = 6, and 20 − 6 = 14.
Answer: 14
EXAMATLAS LEVEL
Q. Simplify 54 ÷ [3 + {6 × (5 − 2) − 3 of 4}] + 2 of 3 ÷ 6.
5 − 2 = 3, so 6 × 3 = 18; 3 of 4 = 12; the brace is 6 and the square bracket 9. 54 ÷ 9 = 6. Then 2 of 3 = 6 and 6 ÷ 6 = 1. Total = 7.
Answer: 7
Pattern 3: Square, cube and fourth roots in an expression
[PYQ: NTPC Graduate CBT-1 5-Jun-2025 Shift-1]
EXAM LEVEL
Q. Simplify √144 + ∛125 − √0.09.
12 + 5 − 0.3 = 16.7.
Answer: 16.7
EXAMATLAS LEVEL
Q. Simplify (∛2744 × ⁴√625) ÷ (√0.0049 × 100).
∛2744 = 14 and ⁴√625 = 5, so the numerator is 70. √0.0049 = 0.07, so the denominator is 7. Value = 10.
Answer: 10
Pattern 4: Decimal products and quotients
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find 0.5 × 0.05 × 0.005.
5 × 5 × 5 = 125 with 1 + 2 + 3 = 6 decimal places: 0.000125.
Answer: 0.000125
EXAMATLAS LEVEL
Q. Simplify (0.08 × 0.6 × 25)/(0.4 × 0.03 × 2).
Numerator = 0.048 × 25 = 1.2. Denominator = 0.012 × 2 = 0.024. 1.2/0.024 = 1,200/24 = 50.
Answer: 50
Pattern 5: Continued fractions
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Simplify 1 + 1/(1 + 1/(1 + 1/2)).
From the bottom: 1 + 1/2 = 3/2, its reciprocal 2/3, then 1 + 2/3 = 5/3, its reciprocal 3/5, and finally 1 + 3/5 = 8/5.
Answer: 8/5
EXAMATLAS LEVEL
Q. Find x if 1/(2 + 1/(3 + 1/x)) = 7/16.
Invert: 2 + 1/(3 + 1/x) = 16/7, so 1/(3 + 1/x) = 2/7 and 3 + 1/x = 7/2. Then 1/x = 1/2 and x = 2.
Answer: x = 2
60-Second Revision
- Brackets, then "of", then ÷ and × left to right, then + and −.
- "Of" comes before ÷.
- Know squares to 30, cubes to 20 and decimal roots.
- Continued fractions: work from the bottom up, or invert step by step.