Percentages & Averages in General AptitudePercentages & Averages in General Aptitude

Core principles of calculating proportions out of a hundred, fractional equivalents, and percentage change formulas essential for RO/ARO prelims.

• Percentage to Fraction Conversion: To convert any percentage to a fraction, divide by 100. For instance, 16.67% equals 1/6, 12.5% equals 1/8, and 37.5% equals 3/8, significantly reducing calculation time.
• Successive Percentage Change: When a value undergoes successive percentage changes of 'a%' and 'b%', the net percentage change is given by the formula: [a + b + (a × b) / 100]%.
• Percentage Point vs Percentage Increase: A rise from 10% to 15% is an absolute increase of 5 percentage points, but a relative percentage increase of (5/10) × 100 = 50%. Exam questions frequently test this distinction.
• Consumption and Price Inverse Rule: If the price of a commodity increases by r%, the reduction in consumption to keep expenditure constant is given by [r / (100 + r)] × 100%.

Understanding arithmetic means, properties of group additions/deletions, and weighted averages crucial for data sufficiency and reasoning puzzles.

• Basic Arithmetic Average: Average = (Sum of all observations) / (Total number of observations). Any change in every observation equally affects the average by the exact same margin.
• Inclusion and Exclusion Principle: When a new entity is included, New Sum = (Old Average × Old Count) + New Value. If the average increases by 'k', the new value equals Old Average + (New Count × k).
• Weighted Average Formula: When groups with different averages are combined, Weighted Average = (n1.A1 + n2.A2) / (n1 + n2), where A represents averages and n represents group sizes.
• Speed Involving Averages (Harmonic Mean): When equal distances are covered at speeds 'x' and 'y', the average speed is NOT (x+y)/2, but the Harmonic Mean: 2xy / (x + y).

Bridging percentages and averages to interpret complex tabular and graphical data sets often featured in Uttar Pradesh administrative examinations.

• Percentage Growth Rate in Data Sets: Growth Rate (%) = [(Value in Later Year - Value in Earlier Year) / Value in Earlier Year] × 100. Always ensure the base year is correctly identified in tables.
• Average Annual Growth Rate (AAGR): Calculated by taking the arithmetic mean of year-on-year percentage growth rates across a specified multi-year timeframe in administrative statistical reports.
• Ratio vs Percentage Comparison: Questions asking 'A is what percent of B' require (A/B) × 100, whereas 'A is what percent more than B' requires [(A - B) / B] × 100.
• Data Sufficiency Checkpoints: In data sufficiency, statement 1 providing total sum and statement 2 providing individual weights are often interdependent to calculate correct group averages.