Simplification

Concepts & Definitions

• Simplification: Reducing a complex mathematical expression to its simplest form by following a fixed order of operations.
• VBODMAS: The order in which operations must be solved: Vinculum, Brackets, Orders (powers, roots) / Of, Division, Multiplication, Addition, Subtraction.
• Regional variations: BODMAS (India/UK), BIDMAS / BEDMAS (UK/Canada) and PEMDAS (USA) follow exactly the same logic.

The VBODMAS Order

Letter	Operation	What to do
V	Vinculum (Bar)	Solve the terms under a horizontal bar first, treating them as one bracket.
B	Brackets	Work from the innermost to the outermost: ( ) → { } → [ ].
O	Orders / Of	Evaluate powers, roots and the word “of” (which means multiplication).
D, M	Division, Multiplication	Equal priority. Solve left to right, in the order they appear.
A, S	Addition, Subtraction	Equal priority. Solve left to right, in the order they appear.

Essential Formulas

 

Algebraic Identities

Identity	Use / Logic
(a + b)² = a² + b² + 2ab	Expand a sum squared; find a² + b² from a + b and ab.
(a − b)² = a² + b² − 2ab	Expand a difference squared; find a² + b² from a − b and ab.
(a + b)² = (a − b)² + 4ab	Convert between sum and difference forms.
a² − b² = (a − b)(a + b)	Quick squares, e.g. 87² − 13² = 74 × 100.
a³ + b³ = (a + b)(a² − ab + b²)	Sum of cubes.
a³ − b³ = (a − b)(a² + ab + b²)	Difference of cubes.
a² + b² = ½ [(a + b)² + (a − b)²]	Find a² + b² when both a + b and a − b are known.

Sums of Counting Numbers

Series	Formula
Sum of first n natural numbers	n(n + 1) / 2
Sum of first n odd natural numbers	n²
Sum of first n even natural numbers	n(n + 1)

Laws of Indices (Exponents)

Law	Example
a&supm; × aⁿ = aᵐ⁺ⁿ	2³ × 2&sup4; = 2⁷
a&supm; ÷ aⁿ = aᵐ⁻ⁿ	3&sup5; ÷ 3² = 3³
(a&supm;)ⁿ = aᵐⁿ	(2³)² = 2⁶
(ab)ⁿ = aⁿ bⁿ	(2 × 3)² = 2² × 3²
(a/b)ⁿ = aⁿ / bⁿ	(2/3)² = 4/9
a⁰ = 1	7⁰ = 1

Laws of Surds

Law	Example
ⁿ√a = a^(1/n)	√9 = 9^(1/2) = 3
ⁿ√(ab) = ⁿ√a × ⁿ√b	√12 = √4 × √3 = 2√3
ⁿ√(a/b) = ⁿ√a / ⁿ√b	√(50/2) = √50 / √2 = 5
(ⁿ√a)ⁿ = a	(√5)² = 5
(ⁿ√a)ᵐ = ⁿ√(aᵐ)	(√3)³ = √27

Bracket Sign Rules

• x + (y + z) = x + y + z
• x − (y + z) = x − y − z (the minus sign changes every sign inside)
• x(y + z) = xy + xz

Essential Shortcuts & Solution Methods

 

A. Direct VBODMAS Evaluation

• Procedure: Clear the innermost bracket or bar first, then orders/of, then division and multiplication (left to right), then addition and subtraction (left to right).
• Example: 16 + (10 − 3 × 2). Bracket first: 10 − 6 = 4. Then 16 + 4 = 20.
• Pitfall 1: Division does not beat multiplication. 8 ÷ 2 × 4 = 4 × 4 = 16 (not 8 ÷ 8 = 1).
• Pitfall 2: Addition does not beat subtraction. 1 − 3 + 4 = −2 + 4 = 2 (not 1 − 7 = −6).

B. Symbol Substitution

• Procedure: Replace each coded symbol with the operation it stands for, rewrite the whole expression, then apply VBODMAS. Substitute all signs at once, never one by one.
• Example: If ‘+’ means ‘÷’, ‘−’ means ‘×’, ‘×’ means ‘+’, ‘÷’ means ‘−’, find 45 + 9 − 3 × 15 ÷ 2.
  • Substitute: 45 ÷ 9 × 3 + 15 − 2 → 5 × 3 + 15 − 2 → 15 + 15 − 2 = 28.

C. Interchanging Signs and Numbers

• Procedure: Swap each pair of signs (or numbers) given in the options, one option at a time, and evaluate the left side. The correct option is the one that makes both sides equal.
• Time saver: First compare the current left-hand value with the right-hand value to see whether you need a bigger or smaller result, then eliminate options that go the wrong way.

D. Clearing Decimals and Fractions in Linear Equations

• Decimals: Multiply both sides by 10, 100, … according to the maximum number of decimal places.
  • 2.3x + 2.8 = −1.2x + 9.8 → (×10) → 23x + 28 = −12x + 98 → 35x = 70 → x = 2.
• Fractions: Multiply both sides by the LCM of all denominators.
  • x/3 + 1/5 = x/5 − 1 → (×15) → 5x + 3 = 3x − 15 → 2x = −18 → x = −9.

E. Repeating Decimals to Fractions

• Pure repeating (0.818181…): Let x = 0.818181…. Two digits repeat, so multiply by 100: 100x = 81.8181… = 81 + x.
  • 99x = 81, so x = 81/99 = 9/11.
• Shortcut: For a pure repeating decimal, the fraction is (repeating digits) / (as many 9s as digits). So 0.(27) = 27/99 = 3/11.

SIMPLIFICATION EXERCISES

 

1. Simplify: √5625 ÷ 15 × 21 ÷ 3 + 5 = ?

(A) 40
(B) 30
(C) 25
(D) 35

Answer: (A) 40

Solution:
√5625 = 75
75 ÷ 15 × 21 ÷ 3 + 5
= 5 × 21 ÷ 3 + 5
= 5 × 7 + 5
= 35 + 5 = 40

2. Evaluate: 15% of 800 + 35 × 4 − 150 ÷ 3 = ?

(A) 200
(B) 210
(C) 220
(D) 230

Answer: (B) 210

Solution:
15% of 800 = 120 ; 35 × 4 = 140 ; 150 ÷ 3 = 50
120 + 140 − 50 = 210

3. Find the value of: (15 × 18) + (255 ÷ 17) − (189 ÷ 9) = ?

(A) 264
(B) 270
(C) 285
(D) 291

Answer: (A) 264

Solution:
15 × 18 = 270; 255 ÷ 17 = 15; 189 ÷ 9 = 21
270 + 15 − 21 = 264

4. Simplify: 5/16 − 1/8 − 3/32 = 3/16 − 1/4 + ?/32

(A) 3
(B) 4
(C) 5
(D) 2

Answer: (C) 5

Solution:
Taking LCM of 16, 8, 4, 32 = 32
(10 − 4 − 3)/32 = (6 − 8 + ?)/32
⇒ 3 = (−2 + ?)
⇒ ? = 5

4. Evaluate: 15 × ? + 20% of 450 = 360 + 15 × 8

(A) 24
(B) 26
(C) 28
(D) 32

Answer: (B) 26

Solution:
20% of 450 = 90 ; 360 + 15 × 8 = 480
15 × ? + 90 = 480
⇒ 15 × ? = 390
⇒ ? = 390/15 = 26

5. Evaluate: 23% of 600 + 33% of 800 − 53% of 400 = ?

(A) 170
(B) 190
(C) 210
(D) 180

Answer: (B) 190

Solution:
23% of 600 = 138; 33% of 800 = 264; 53% of 400 = 212
138 + 264 − 212 = 190

6. Find the value of: 56% of 350 + 48% of 550 − 15 × 2.4 = ?

(A) 412
(B) 424
(C) 436
(D) 448

Answer: (B) 424

Solution:
56% of 350 = 196; 48% of 550 = 264; 15 × 2.4 = 36
196 + 264 − 36 = 424

7. Simplify: √196 × √144 × 20% of 500 = ? + 1256

(A) 15544
(B) 15572
(C) 15589
(D) 15511

Answer: (A) 15544

Solution:
√196 = 14; √144 = 12; 20% of 500 = 100
14 × 12 × 100 = ? + 1256
⇒ ? = 16800 − 1256 = 15544

8. If 48% of ? + 45% of 600 = 19² + 841, find the approximate value of ?.(A) 1950
(B) 1941
(C) 1940
(D) 1942

Answer: (D) 1942

Solution:
45% of 600 = 270 ; 19² + 841 = 361 + 841 = 1202
48% of ? = 1202 − 270 = 932
⇒ ? = (932 × 100)/48 ≈ 1942

9. Evaluate: 17² + 6 × 13² − 212 × 6 = ?

(A) 25
(B) 31
(C) 37
(D) 41

Answer: (B) 31

Solution:
17² = 289; 13² = 169
289 + 6 × 169 − 212 × 6
= 289 + 1014 − 1272
= 1303 − 1272 = 31

10. Evaluate: 8% of 125 × ?% of 120 = 420

(A) 35
(B) 30
(C) 25
(D) 40

Answer: (A) 35

Solution:
8% of 125 = 10
10 × ?% of 120 = 420
⇒ ?% of 120 = 42
⇒ ? = (42 × 100)/120 = 35

11. Simplify: 25% of 680 + 16% of 525 − 22% of 450 = ?

(A) 145
(B) 155
(C) 165
(D) 175

Answer: (B) 155

Solution:
25% of 680 = 170 ; 16% of 525 = 84 ; 22% of 450 = 99
170 + 84 − 99 = 155

12. Find the value of: √784 × √6.25 + 56% of 450 − 186 = ?

(A) 136
(B) 121
(C) 107
(D) 114

Answer: (A) 136

Solution:
√784 = 28 ; √6.25 = 2.5 ; 56% of 450 = 252
28 × 2.5 + 252 − 186
= 70 + 252 − 186 = 136

13. Evaluate: 12 × 15 + 27 − 28% of 400 = ?

(A) 95
(B) 105
(C) 115
(D) 125

Answer: (A) 95

Solution:
12 × 15 = 180 ; 28% of 400 = 112
180 + 27 − 112 = 95

14. Evaluate: √576 ÷ (4)² × 7.4 + (7)³ − 231 = ?

(A) 115
(B) 123
(C) 120.1
(D) 123.1

Answer: (D) 123.1

Solution:
√576 = 24 ; (4)² = 16 ; (7)³ = 343
24 ÷ 16 × 7.4 + 343 − 231
= 1.5 × 7.4 + 112
= 11.1 + 112 = 123.1