Express $ 0.\overline{52}$ in the form $\frac{p}{q}$, where $ ... | Mathematics MCQ | Examoogle

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Express $0.\overline{52}$ in the form $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ .

$\frac{52}{90}$

$\frac{53}{99}$

$\frac{50}{99}$

$\frac{52}{99}$

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option D

$\frac{52}{99}$

Quick Summary:

For a pure repeating decimal of the form $0.\overline{ab}$ , the equivalent fraction is $\frac{ab}{99}$ . Here, $0.\overline{52}$ becomes $\frac{52}{99}$ .

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📐MAMath SolutionDirect Formula

📋 Given

The repeating decimal is 0.52 (with a bar over 52), representing 0.525252...

🔢 Formula Used

x=0.p‾=p99...9 (where 9’s correspond to the length of the recurring period)x = 0.\overline{p} = \frac{p}{99...9} \text{ (where 9's correspond to the length of the recurring period)}x=0.p​=99...9p​ (where 9’s correspond to the length of the recurring period)

⚡ Exam Hall Shortcut / Speed Trick

For a pure repeating decimal of the form $0.\overline{ab}$ , the equivalent fraction is $\frac{ab}{99}$ . Here, $0.\overline{52}$ becomes $\frac{52}{99}$ .

⚠️ Common Student Trap / Pitfall

Students often mistakenly divide by 90 (which is used for mixed recurring decimals like $0.a\overline{b}$ ) instead of 99.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Define the equation

Let $x = 0.\overline{52} = 0.525252...$

x=0.525252...x = 0.525252...x=0.525252...

Shift the decimal

Since two digits repeat, multiply by $10^2 = 100$ to shift the decimal point two places to the right:

100x=52.525252...100x = 52.525252...100x=52.525252...

Subtract the equations

Subtract the original equation $x$ from $100x$ to eliminate the repeating part:

100x−x=52.525252...−0.525252...100x - x = 52.525252... - 0.525252...100x−x=52.525252...−0.525252...

Solve for x

Simplify the resulting expression $99x = 52$ to find the fraction:

x=5299x = \frac{52}{99}x=9952​

✅

D is correct because applying the standard rule for converting pure recurring decimals to fractions, $0.\overline{52} = \frac{52}{99}$ .

Core Concepts Used

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Recurring Decimals Rational Numbers Algebraic Manipulation

💡 EXAM TIP

This concept is foundational for Number System problems. Remember that for any decimal with a bar over $n$ digits, the denominator is a string of $n$ nines.

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