Simplification: Notes और MCQ Practice

आसान भाषा में:

Simplification का मतलब है बड़े और डरावने दिखने वाले mathematical expressions को आसान बनाना। SSC CGL Exam में इससे direct 2-3 questions पूछे जाते हैं। बस VBODMAS rule और basic algebraic formulas का सही use करना है और marks आपकी जेब में!

Important Rules & Formulas:

• VBODMAS Rule: Order of operations हमेशा याद रखें:
• V (Vinculum/Bar: ‾)
• B (Brackets: (), {}, [])
• O (Of: गुणा, but Division से पहले)
• D (Division: ÷)
• M (Multiplication: ×)
• A (Addition: +)
• S (Subtraction: -)
• Algebraic Identities:
• a² - b² = (a + b)(a - b)
• a³ + b³ = (a + b)(a² - ab + b²)
• (a + b)² = a² + 2ab + b²

2 Quick Tricks:

1. Digital Sum Method: जब calculation बहुत लंबी हो, तो options और question के digits का sum match करें।
2. Unit Digit Technique: पूरा solve करने के बजाय सिर्फ unit digits को calculate करके correct option select करें।

2 Common Mistakes:

1. 'Of' vs '÷' की गलती: 'Of' का calculation हमेशा Division (÷) से पहले होता है। Example: 10 ÷ 5 of 2 = 10 ÷ 10 = 1 (न कि 2 × 2 = 4).
2. Sign Change Error: Bracket के बाहर '-' sign होने पर, bracket open करते ही अंदर के सारे signs change हो जाते हैं। इसे भूलना नहीं है!

Simplification Study Notes

Simplification is one of the most high-yielding and scoring topics in the Quantitative Aptitude section of the SSC CGL exam. Usually, 2 to 3 questions are asked directly from this topic in Tier-1, and they can be solved quickly if you know the basic rules and calculation tricks.

1. Concept of Simplification

At its core, simplification means reducing a complex mathematical expression containing multiple numbers and operations (+, −, ×, ÷, brackets, etc.) into a single, simple value.

To solve these expressions correctly, you must follow a strict order of operations. In mathematics, this order is defined by the VBODMAS rule. Without this rule, different people would get different answers for the same expression.

2. Important Formulas, Rules, and Facts

The VBODMAS Rule

You must solve any expression from left to right in the exact order of the letters in VBODMAS:

• V (Vinculum or Bar Bracket): Represented as a line over numbers (e.g., 5 − 3 with a bar over it). Solve this first.
• B (Brackets): Solve brackets in this specific order:

1. Small brackets or Parentheses: ( )
2. Curly brackets or Braces: { }
3. Square brackets or Box brackets: [ ]

• O (Of): "Of" represents multiplication (e.g., 5 of 10 = 5 × 10 = 50). Note: This is solved before division.
• D (Division): Perform division (÷).
• M (Multiplication): Perform multiplication (×).
• A (Addition): Perform addition (+).
• S (Subtraction): Perform subtraction (−).

Crucial Algebraic Identities

Many simplification questions are secretly algebraic identities in disguise. Memorize these to save time:

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• a² − b² = (a − b)(a + b)
• a³ + b³ = (a + b)(a² − ab + b²)
• a³ − b³ = (a − b)(a² + ab + b²)
• If a + b + c = 0, then a³ + b³ + c³ = 3abc

Rules of Surds and Indices

• a^m × a^n = a^(m + n)
• a^m ÷ a^n = a^(m − n)
• (a^m)^n = a^(m × n)
• a^0 = 1
• a^(−n) = 1 ÷ a^n

3. Solved Examples (Exam Style)

Example 1: Basic VBODMAS

Question: Simplify the expression: 1800 ÷ 10 of 18 + (15 × 10 − (20 − 5))

Step-by-step Solution:

1. Step 1: Solve the innermost small bracket first: (20 − 5) = 15.

• Expression becomes: 1800 ÷ 10 of 18 + (15 × 10 − 15)

1. Step 2: Solve the remaining bracket. Inside it, perform multiplication before subtraction:

• (150 − 15) = 135
• Expression becomes: 1800 ÷ 10 of 18 + 135

1. Step 3: Solve the "of" operation (10 of 18 = 10 × 18 = 180):

• Expression becomes: 1800 ÷ 180 + 135

1. Step 4: Perform division (1800 ÷ 180 = 10):

• Expression becomes: 10 + 135

1. Step 5: Perform addition:

• 10 + 135 = 145

Answer: 145

Example 2: Algebraic Identity Based

Question: Find the value of: (0.87 × 0.87 × 0.87 + 0.13 × 0.13 × 0.13) ÷ (0.87 × 0.87 − 0.87 × 0.13 + 0.13 × 0.13)

Step-by-step Solution:

1. Step 1: Identify the pattern. Let a = 0.87 and b = 0.13.
2. Step 2: Rewrite the expression using variables:

• Numerator = a³ + b³
• Denominator = a² − ab + b²
• Expression = (a³ + b³) ÷ (a² − ab + b²)

1. Step 3: Apply the algebraic identity: a³ + b³ = (a + b)(a² − ab + b²).

• Expression = [(a + b)(a² − ab + b²)] ÷ (a² − ab + b²)

1. Step 4: Cancel out the common term (a² − ab + b²) from numerator and denominator:

• Remaining expression = a + b

1. Step 5: Substitute the values back:

• 0.87 + 0.13 = 1.00

Answer: 1

Example 3: Infinite Nested Roots

Question: Find the value of: √[12 + √{12 + √{12 + ... to infinity}}]

Step-by-step Solution:

1. Step 1: Let the entire expression be equal to x:

• x = √[12 + √{12 + √{12 + ...}}]

1. Step 2: Since the expression goes to infinity, we can replace the nested part with x:

• x = √[12 + x]

1. Step 3: Square both sides to remove the outer square root:

• x² = 12 + x

1. Step 4: Rearrange into a quadratic equation:

• x² − x − 12 = 0

1. Step 5: Factorize the equation:

• x² − 4x + 3x − 12 = 0
• x(x − 4) + 3(x − 4) = 0
• (x − 4)(x + 3) = 0

1. Step 6: Solve for x. Since a square root value cannot be negative, we discard x = −3.

• x = 4

Answer: 4

4. Shortcuts and Tricks

Trick 1: Infinite Root Shortcut

For questions like √[N + √{N + √{N + ...}}]:

• Find two consecutive factors of N (i.e., n and n + 1) such that their product is N.
• If there is a plus (+) sign, the answer is the larger factor (n + 1).
• If there is a minus (−) sign, the answer is the smaller factor (n).
• Example: For √[12 + √{12 + ...}], factors of 12 are 3 × 4. Since it is '+', the answer is 4.

Trick 2: Digital Sum Method

When options have different digit sums, you can avoid full calculations:

• Find the single-digit sum of the numbers in the question by adding their digits (replace any 9 or sum adding to 9 with 0).
• Perform the operations on these single digits.
• Match this final digital sum with the digital sums of the options.

Trick 3: Unit Digit Concept

If the options in the exam have different unit digits, do not calculate the entire expression. Only perform the operations on the unit digits of the numbers to find the last digit of the answer.

5. Common Mistakes to Avoid

1. Confusing "Of" with "Multiplication (×)": Always remember that "Of" is solved before division, whereas "×" is solved after division.

• Incorrect: 10 ÷ 2 of 5 = 10 ÷ 2 × 5 = 5 × 5 = 25
• Correct: 10 ÷ 2 of 5 = 10 ÷ (2 × 5) = 10 ÷ 10 = 1

1. Sign Mistakes with Brackets: When opening a bracket with a minus (−) sign outside, remember to change the signs of all terms inside the bracket.

• −(a − b) = −a + b

1. Forgetting the Vinculum (Bar): If a bar is present over an expression, it must be solved first, even before the small brackets. For example, in 5 − 3 − 2 with a bar over 3 − 2, you must solve 3 − 2 first.