Percentage: Notes और MCQ Practice

आसान भाषा में:

Percentage का मतलब है "per 100" यानी हर 100 पर कितना। SSC CGL में Percentage पूरे Arithmetic का base है। अगर आपने Percentage to Fraction conversion और Base Value का concept समझ लिया, तो Profit & Loss, SI-CI, और DI के सवाल चुटकियों में हल हो जाएंगे।

Important Formulas & Rules:

• Fraction to Percentage: Fraction × 100 (जैसे: 1/5 × 100 = 20%)
• Percentage to Fraction: Percentage ÷ 100 (जैसे: 25% = 25 ÷ 100 = 1/4)
• Percentage Change = (Change ÷ Original Value) × 100
• Successive Percentage Change = A + B + (A × B) ÷ 100 (Increase के लिए '+' और Decrease के लिए '-' sign use करें)

2 Quick Tricks:

1. x% of y = y% of x: Calculation आसान करने के लिए 64% of 50 निकालने के बजाय 50% of 64 निकालो, answer 32 ही आएगा!
2. Product Constancy (AB Rule): अगर Price, x% बढ़ता है और Expenditure constant रखना है, तो Consumption में reduction = [x ÷ (100 + x)] × 100% होगा।

2 Common Mistakes to Avoid:

1. Wrong Base Value: Percentage increase/decrease निकालते समय हमेशा denominator में Original (Initial) value रखें, New value नहीं।
2. Direct Addition in Successive Change: अगर population पहले साल 10% और दूसरे साल 20% बढ़ती है, तो total change 30% नहीं, बल्कि Successive formula से 32% होगा।

Study Notes: Percentage (Quantitative Aptitude)

This guide is designed for SSC CGL aspirants, especially those who find the concept of percentages challenging. By the end of these notes, you will understand the core concepts, remember key formulas, and learn tricks to solve exam questions quickly.

1. Concept of Percentage

The word "Percent" comes from the Latin "per centum", which means "per hundred" or "out of 100".

• Represented by the symbol %.
• A percentage is simply a fraction whose denominator is always 100.
• For example, 25% means 25 out of 100, which can be written as the fraction 25 ÷ 100 = 1 ÷ 4.

Conversion Rules:

1. To convert a percentage into a fraction: Divide the percentage value by 100.

• Example: 40% = 40 ÷ 100 = 2 ÷ 5

1. To convert a fraction into a percentage: Multiply the fraction by 100.

• Example: 3 ÷ 4 = (3 ÷ 4) × 100% = 75%

2. Important Formulas, Rules, and Facts

Basic Formulas

• Percentage Value = (Part Value ÷ Total Value) × 100
• Percentage Increase = (Increase in Value ÷ Original Value) × 100
• Percentage Decrease = (Decrease in Value ÷ Original Value) × 100

Successive Percentage Change

If a value is changed successively by A% and then by B%, the net percentage change is given by the formula:

• Net Percentage Change = [A + B + (A × B) ÷ 100]%
• Note: Use a positive (+) sign for an increase/profit and a negative (−) sign for a decrease/loss.

Product Constancy Rule (AB Rule)

If the product of two quantities is constant (e.g., Price × Consumption = Expenditure):

• If the price of a commodity increases by r%, then the reduction in consumption required to keep the expenditure unchanged is:
• Reduction % = [r ÷ (100 + r)] × 100%
• If the price decreases by r%, the increase in consumption required to keep the expenditure unchanged is:
• Increase % = [r ÷ (100 − r)] × 100%

Essential Fraction-to-Percentage Conversion Table

Memorizing these conversions will save you valuable time during the exam:

• 1 = 100%
• 1 ÷ 2 = 50%
• 1 ÷ 3 = 33.33% (or 33 1/3%)
• 1 ÷ 4 = 25%
• 1 ÷ 5 = 20%
• 1 ÷ 6 = 16.66% (or 16 2/3%)
• 1 ÷ 7 = 14.28% (or 14 2/7%)
• 1 ÷ 8 = 12.5% (or 12 1/2%)
• 1 ÷ 9 = 11.11% (or 11 1/9%)
• 1 ÷ 10 = 10%
• 1 ÷ 11 = 9.09% (or 9 1/11%)
• 1 ÷ 12 = 8.33% (or 8 1/3%)

3. Solved Examples (Exam Style)

Example 1: Income Comparison

Question: If A's salary is 25% more than B's salary, by what percentage is B's salary less than A's salary?

Step-by-Step Solution:

1. Let B's salary be 100.
2. Since A's salary is 25% more than B's, A's salary = 100 + 25% of 100 = 125.
3. We need to find how much B's salary is less than A's. The difference is 125 − 100 = 25.
4. The comparison is with A's salary, so A's salary (125) becomes the base (denominator).
5. Required Percentage = (Difference ÷ A's Salary) × 100

• = (25 ÷ 125) × 100
• = (1 ÷ 5) × 100 = 20%

Answer: B's salary is 20% less than A's.

Example 2: Successive Change

Question: The price of a mobile phone is first increased by 20% and then decreased by 10% due to a sale. What is the net percentage change in the price of the mobile phone?

Step-by-Step Solution:

• Method 1: Using the Successive Formula
• Here, Increase (A) = +20%
• Decrease (B) = −10%
• Net Change = [A + B + (A × B) ÷ 100]%
• Net Change = [20 + (−10) + (20 × −10) ÷ 100]%
• Net Change = [10 − 200 ÷ 100]%
• Net Change = 10 − 2 = 8%
• Since the result is positive, there is an 8% net increase.

• Method 2: Assuming Base 100
• Let the original price = 100.
• After 20% increase, price = 120.
• Now, decrease this new price by 10%: 10% of 120 = 12.
• New price = 120 − 12 = 108.
• Net change from original (100 to 108) = 8% increase.

Answer: The net price increased by 8%.

Example 3: Election-Based Problem

Question: In an election between two candidates, the winning candidate got 60% of the total valid votes and won by a majority of 12,000 votes. Find the total number of valid votes cast.

Step-by-Step Solution:

1. Let the total number of valid votes be 100%.
2. The winning candidate gets 60% of the votes.
3. Therefore, the losing candidate gets: 100% − 60% = 40% of the votes.
4. The margin of victory (difference in percentage) = 60% − 40% = 20%.
5. According to the question, this 20% difference is equal to 12,000 votes.

• 20% of Total Votes = 12,000
• 1% of Total Votes = 12,000 ÷ 20 = 600
• 100% of Total Votes = 600 × 100 = 60,000

Answer: The total number of valid votes is 60,000.

4. Shortcuts and Tricks

Trick 1: The Fraction Method

Instead of using large numbers, convert percentages to fractions to make calculations faster.

• Example: If a number is increased by 16.66%, it means it is increased by 1 ÷ 6 of its value.
• This means the original value was 6 units, the increase is 1 unit, and the new value is 7 units.

Trick 2: Multiplying Factor (MF)

To find the final value quickly, multiply the original value by its Multiplying Factor:

• For an increase of x%, multiply by (1 + x ÷ 100).
• Example: A 15% increase means multiplying by 1.15.
• For a decrease of x%, multiply by (1 − x ÷ 100).
• Example: A 20% decrease means multiplying by 0.80.

Trick 3: Solving Price-Consumption Problems Instantly

If the price of sugar increases by 25%, by how much must a family reduce consumption to keep expenditure the same?

• 25% as a fraction is 1 ÷ 4.
• Since the price increases, add the numerator to the denominator for the reduction fraction:
• New fraction = Numerator ÷ (Denominator + Numerator) = 1 ÷ (4 + 1) = 1 ÷ 5.
• Convert 1 ÷ 5 to percentage: (1 ÷ 5) × 100% = 20% reduction.

5. Common Mistakes to Avoid

• Mistake 1: Confusing the Base Value (Denominator)
• Error: Always pay attention to what the question is comparing a value to. In "A is what percent of B", the base is B. In "B is what percent of A", the base is A.

• Mistake 2: Adding Successive Percentages Directly
• Error: If a population increases by 10% in the first year and 10% in the second year, students often think the total increase is 20%.
• Correction: It is a successive change. The actual increase is 10 + 10 + (10 × 10) ÷ 100 = 21%.

• Mistake 3: Misinterpreting "Reduced to" vs. "Reduced by"
• "Reduced by 20%" means the new value is 80% of the original value.
• "Reduced to 20%" means the new value is exactly 20% of the original value.