In-Depth: Percentages

CLAT Application & Relevance

Importance: VERY HIGH. Percentages are the language of Data Interpretation. They are used to measure growth, decline, contribution, and comparison. Mastering the nuances of percentage calculations is non-negotiable for a good score in the QT section.

How it's tested: Calculating percentage increase/decrease in a company's profit shown on a line graph; finding the percentage contribution of an expense in a pie chart; calculating percentage change in population from a data table.

Section 1: Core Concepts & Formulas

A percentage is a way of expressing a number as a fraction of 100. It removes the base and allows for standardized comparison.

Fundamental Formulas

Concept of 'per cent': x% = x/100
Finding a percentage of a number: To find x% of Y, calculate (x/100) * Y
Expressing X as a percentage of Y: (X / Y) * 100. (The value that comes after "of" is always the base/denominator).

Formulas for Change & Comparison

Percentage Change (Increase/Decrease): ((Final Value - Initial Value) / Initial Value) * 100. This is the most frequently used formula in CLAT.
Finding Final Value after % change:
- After x% increase: Final Value = Initial Value * (1 + x/100) or Initial Value * ((100+x)/100)
- After x% decrease: Final Value = Initial Value * (1 - x/100) or Initial Value * ((100-x)/100)
Finding Initial Value from Final Value:
- If Final Value is known after x% increase: Initial Value = Final Value / (1 + x/100)
- If Final Value is known after x% decrease: Initial Value = Final Value / (1 - x/100)
Percentage Point Change: This is the simple arithmetic difference between two percentage values. E.g., if a company's market share changes from 15% to 18%, the increase is 3 percentage points.

Formulas for Advanced Applications

Successive Percentage Change: If a value is first changed by A% and then by B%, the net percentage change is: A + B + (A*B)/100. (Use '+' for increase, '-' for decrease. For instance, a 20% increase followed by a 10% decrease is +20 - 10 + (20 * -10)/100 = 10 - 2 = 8% net increase).
Product Constancy Rule: If Price × Consumption = Expenditure, and the price increases by x%, the consumption must be decreased by (x / (100+x)) * 100% to keep the expenditure constant.

Section 2: Solved CLAT-Style Examples

Example 1: Basic Percentage Change

Passage Context: A data table shows the number of environmental law cases filed in a district court was 250 in 2022. The number increased to 325 in 2023.

Question: "By what percentage did the number of cases increase from 2022 to 2023?"

Detailed Solution:
1. Identify Initial and Final Values: Initial Value (2022) = 250. Final Value (2023) = 325.
2. Calculate the Absolute Change: Increase = 325 - 250 = 75.
3. Apply the Percentage Change Formula: (Change / Initial Value) * 100
= (75 / 250) * 100
4. Simplify and Calculate: (3 / 10) * 100 = 30%.
Answer: The number of cases increased by 30%.

Example 2: Reverse Percentage Calculation

Passage Context: A pie chart shows that a law firm spent 35% of its annual budget on associate salaries. A footnote mentions that this salary expenditure amounted to ₹70 Lakhs.

Question: "What is the total annual budget of the law firm?"

Detailed Solution:
1. Set up the equation: Let the total budget be 'B'. We are given that 35% of B is ₹70 Lakhs.
2. Formulate the algebra: (35 / 100) * B = 70
3. Solve for B: B = 70 * (100 / 35)
4. Calculate: B = 2 * 100 = 200.
Answer: The total annual budget of the firm is ₹200 Lakhs.

Example 3: Successive Percentage Change

Passage Context: Due to a new policy, the number of applications for a particular corporate license first increased by 20% in the first quarter of the year. However, due to market saturation, the number of applications then decreased by 20% in the second quarter.

Question: "What is the net percentage change in the number of applications after the two quarters compared to the beginning of the year?"

Detailed Solution:
1. Identify A and B: The first change (A) is an increase, so A = +20. The second change (B) is a decrease, so B = -20.
2. Apply the Successive Change Formula: A + B + (A*B)/100
= (+20) + (-20) + ((+20) * (-20)) / 100
3. Calculate: = 0 + (-400 / 100) = -4.
Answer: There is a net decrease of 4% in the number of applications. (This is a common trap; it's not a 0% change).