1 General Mental and Psychological Abilities · 1.2.1 Number system and simplification
1.2.1.4 Simplification and approximation: BODMAS, surds, indices and roots
Simplification questions test order of operations, a handful of identities and the habit of spotting a pattern; most collapse to a small whole number once the pattern is seen. (Decimals, roots, and surds and indices have fuller notes: MAB-NS-06, 07 and 08.)
- APPSC Group 1
- Advanced level
- 6 min read
- Free
Core facts
- BODMAS: Brackets → Of → Division and Multiplication (left to right) → Addition and Subtraction (left to right). "Of" means multiply.
- Brackets open from the inside: ( ) first, then { }, then [ ]. A bar (vinculum) over terms acts as a bracket.
- Identities:
- (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a + b)(a − b)
- (a + b)² − (a − b)² = 4ab; (a + b)² + (a − b)² = 2(a² + b²)
- a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²)
- If a + b + c = 0, then a³ + b³ + c³ = 3abc.
- Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ.
- Rationalising: 1/(√a − √b) = (√a + √b)/(a − b).
- Series sums: 1 + 2 + … + n = n(n+1)/2; 1² + … + n² = n(n+1)(2n+1)/6; 1³ + … + n³ = [n(n+1)/2]².
Example 1. Simplify 48 ÷ 6 × 2 + 3 of 4 − 5. Solution. "Of" first: 3 of 4 = 12. Then left to right: 48 ÷ 6 = 8, × 2 = 16. 16 + 12 − 5. Answer: 23
Example 2. Simplify 20 − [8 − {6 − (4 − 1)}]. Solution. (4 − 1) = 3; {6 − 3} = 3; [8 − 3] = 5; 20 − 5. Answer: 15
Example 3. Find 103 × 97 without long multiplication. Solution. (100 + 3)(100 − 3) = 10,000 − 9. Answer: 9,991
Explained
Type 1: identity in disguise. Large numbers placed symmetrically almost always fit an identity.
Example 4. Value of [(467 + 133)² − (467 − 133)²] ÷ (467 × 133). (a) 2 (b) 4 (c) 600 (d) 334 Solution. (a + b)² − (a − b)² = 4ab, so the quotient is 4. Answer: (b) 4
Example 5. Value of (0.7³ + 0.3³) ÷ (0.7² − 0.7 × 0.3 + 0.3²). Solution. (a³ + b³)/(a² − ab + b²) = a + b = 0.7 + 0.3. Answer: 1
Example 6. Value of 17³ − 9³ − 8³. Solution. Take a = 17, b = −9, c = −8; a + b + c = 0. So the value = 3abc = 3 × 17 × (−9) × (−8). Answer: 3,672
Type 2: find the missing number. Simplify each side separately, then solve.
Example 7. 25% of 480 + ? = 3/5 of 350. Solution. 120 + ? = 210. Answer: 90
Type 3: approximation. Round each number to an easy value; options are usually far apart.
Example 8. 39.97% of 601.8 + 14.02 × 9.98 ≈ ? (a) 320 (b) 350 (c) 380 (d) 420 Solution. 40% of 600 = 240; 14 × 10 = 140; 240 + 140 = 380. (Exact value 380.46.) Answer: (c) 380
Type 4: telescoping sums and products. Split each term so that neighbours cancel.
Example 9. 1/(2×3) + 1/(3×4) + … + 1/(29×30). Solution. 1/(n(n+1)) = 1/n − 1/(n+1). Sum = 1/2 − 1/30 = 14/30. Answer: 7/15
Example 10. (1 − 1/3)(1 − 1/4)(1 − 1/5) … (1 − 1/40). Solution. = (2/3)(3/4)(4/5) … (39/40). Everything cancels except 2/40. Answer: 1/20
Speed tips.
- Before calculating, look for a², b², 2ab or a³, b³ patterns; the numbers are chosen to fit one.
- In approximation questions, check how far apart the options are before deciding how roughly to round.
- Division and multiplication go left to right; never do × before ÷ just because "M" is in the word.
Deeper
Example 11. 1/(1×4) + 1/(4×7) + 1/(7×10) + … + 1/(28×31). Solution. Here the factors differ by 3: 1/(n(n+3)) = (1/3)(1/n − 1/(n+3)). Sum = (1/3)(1 − 1/31) = (1/3)(30/31). Answer: 10/31
Example 12. (i) 1³ + 2³ + … + 15³. (ii) 6³ + 7³ + 8³ + 9³ + 10³. Solution. (i) [15 × 16/2]² = 120². (ii) Sum to 10 minus sum to 5: 55² − 15² = 3,025 − 225. Answer: (i) 14,400 (ii) 2,800
Example 13. Simplify 1 + 1/(1 + 1/(1 + 1/2)). Solution. Start at the bottom: 1 + 1/2 = 3/2; 1 + 2/3 = 5/3; 1 + 3/5. Answer: 8/5
Example 14. If x + 1/x = 5, find x³ + 1/x³. (a) 110 (b) 115 (c) 120 (d) 125 Solution. x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x) = 125 − 15. Answer: (a) 110
- Companion: x² + 1/x² = (x + 1/x)² − 2 = 23.
Example 15. Which is the largest: 7/9, 11/14, 15/19, 19/24? Solution. Each fraction is 1 minus something: 2/9, 3/14, 4/19, 5/24. The smallest gap is the largest fraction; 5/24 ≈ 0.208 is the smallest of 0.222, 0.214, 0.211, 0.208. Answer: 19/24
- Other methods: cross-multiply two at a time (a/b > c/d when ad > bc), or divide to two decimal places.
- Sum of squares in a range: 21² + … + 30² = 9,455 − 2,870 = 6,585 (sum to 30 minus sum to 20).
- Checking by substitution: put √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 into the original expression and compare with the options.
Common traps
- ✗ "8 ÷ 2 × 4 = 1 because multiplication comes first." ✓ Division and multiplication go left to right: (8 ÷ 2) × 4 = 16.
- ✗ "x² + 1/x² = (x + 1/x)²." ✓ Subtract 2: (x + 1/x)² − 2.
- ✗ "x³ + 1/x³ = (x + 1/x)³." ✓ Subtract 3(x + 1/x).
- ✗ "√(a + b) = √a + √b." ✓ That holds only for products: √(ab) = √a × √b.
- ✗ "1/(1×4) + … telescopes to 1 − 1/31." ✓ Divide by the gap 3: (1/3)(1 − 1/31).
- ✗ "a⁰ = 0." ✓ a⁰ = 1 for any non-zero a.
Quick revision
- BODMAS; "of" before ÷; ÷ and × left to right.
- (a + b)² − (a − b)² = 4ab.
- a + b + c = 0 ⇒ a³ + b³ + c³ = 3abc.
- x² + 1/x² = (x + 1/x)² − 2; x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x).
- Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².
- 1/(n(n+k)) = (1/k)(1/n − 1/(n+k)).
- Products like (1 − 1/n) telescope to first numerator ÷ last denominator.
- Continued fractions: simplify from the bottom up.
- Compare fractions close to 1 by their gap from 1.
- Approximate only as roughly as the option gaps allow.
Practise this topic. Questions on Clocks, calendars, ages and the number system, with answers explained.
All Number system and simplification notes · APPSC Group 1 syllabus · APPSC G1 Paper I Mock 1 with answers · Start free
Medha AI writes its own notes and checks them against the sources listed. Medha AI is an independent preparation service. It is not affiliated with, endorsed by or connected to APPSC, TSPSC, MPSC, UPSC or any other commission or government body.