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