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MathematicsPrevious Year Paper 2024

Due to decrease in manpower, the production in a factory decreases by 25%. By what percentage should the working hours be increased to restore the original production?

$50\%$

$25\%$

$33\frac{1}{3}\%$

$66\frac{2}{3}\%$

Correct Answer

📐 MA • Math Percentage Change FormulaMathematicsPrevious Year Paper 2024

Option C

$33\frac{1}{3}\%$

Quick Summary:

Use the formula $\frac{x}{100-x} \times 100$ where $x$ is the percentage decrease. For $x=25$ , this is $\frac{25}{75} \times 100 = \frac{1}{3} \times 100 = 33 \frac{1}{3} \%$

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📐MAMath SolutionPercentage Change Formula

📋 Given

Percentage decrease in production is 25%.

🔢 Formula Used

Percentage Increase=(r100−r)×100\text{Percentage Increase} = \left( \frac{r}{100 - r} \right) \times 100Percentage Increase=(100−rr​)×100

⚡ Exam Hall Shortcut / Speed Trick

Use the formula $\frac{x}{100-x} \times 100$ where $x$ is the percentage decrease. For $x=25$ , this is $\frac{25}{75} \times 100 = \frac{1}{3} \times 100 = 33 \frac{1}{3} \%$

⚠️ Common Student Trap / Pitfall

Many students mistakenly apply a 25% increase, assuming the change is symmetric, resulting in a common error of 25%.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Define the relationship

Let initial production be 100 units. A 25% decrease means the new production is $100 - 25 = 75$ units.

Pnew=75P_{new} = 75Pnew​=75

Calculate required increase

To restore production to 100 units, we need to increase the 75 units back to 100. The required increase is $100 - 75 = 25$ units.

Required Increase=25\text{Required Increase} = 25Required Increase=25

Calculate percentage increase

The percentage increase needed is based on the current reduced value (75).

Percentage=(2575)×100=13×100=3313%\text{Percentage} = \left( \frac{25}{75} \right) \times 100 = \frac{1}{3} \times 100 = 33 \frac{1}{3} \%Percentage=(7525​)×100=31​×100=3331​%

✅

D is correct because the required percentage increase to restore the original value after a 25% decrease is calculated as $33 \frac{1}{3} \%$ .

Core Concepts Used

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Percentage change Fractional equivalents Restoration of original value

💡 EXAM TIP

This concept is identical to the 'Price-Consumption' rule in profit/loss and percentage problems: if price increases by $x\%$ , consumption must decrease by $x/(100+x)$ to keep expenditure constant.

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