Correct Answer

📐 MA • Math Direct FormulaMathematicsMensuration

Option C

21%

Quick Summary:

Use the formula $x + y + \frac{xy}{100}$ where $x$ and $y$ are the percentage changes in dimensions; since area depends on $r^2$ , take $x = y = 10\%$ .

📐MAMath SolutionDirect Formula

📋 Given

The radius (r) of a circle is increased by 10%.

🔢 Formula Used

Successive% change=x+y+xy100\text{Successive} \% \text{ change} = x + y + \frac{xy}{100}Successive% change=x+y+100xy​

⚡ Exam Hall Shortcut / Speed Trick

Use the formula $x + y + \frac{xy}{100}$ where $x$ and $y$ are the percentage changes in dimensions; since area depends on $r^2$ , take $x = y = 10\%$ .

⚠️ Common Student Trap / Pitfall

Many students incorrectly assume a 10% increase in radius leads to a 10% increase in area, forgetting that area is a two-dimensional quantity proportional to the square of the radius.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Define the relationship

The area of a circle is given by $A = \pi r^2$ . Because area is proportional to the square of the radius, a change in radius affects the area twice.

A∝r2A \propto r^2A∝r2

Apply Successive Percentage Formula

For any quantity $Q = a \times b$ , if both $a$ and $b$ increase by $x\%$ , the total percentage increase is $x + x + \frac{x^2}{100}$ . Here $x = 10$ .

10+10+10×1010010 + 10 + \frac{10 \times 10}{100}10+10+10010×10​

Calculate the result

Summing the terms gives $20 + \frac{100}{100} = 20 + 1 = 21$ .

21%21\%21%

✅

C is correct because applying the successive percentage formula for a 10% increase in both radius dimensions results in a net increase of 21%.

Core Concepts Used

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Mensuration of Circles Successive Percentage Change Proportionality in Area

💡 EXAM TIP

This same 'successive increase' logic applies to 2D figures like rectangles (length and breadth) and even compound interest problems over two time periods.

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