Question

Express $0.13\overline{241}$ in the form $\frac{p}{q}$, where p and q are integers and $q \neq 0$.

• A. $\frac{3807}{24975}$
• B. $\frac{3287}{21780}$
• C. $\frac{3337}{24160}$
• D. $\frac{3307}{24975}$

Answer 1

Answer:

$\frac{3307}{24975}$

The expression $0.13\overline{241}$ in the form $\frac{p}{q}$ is $\frac{3307}{24975}$

For any mixed recurring decimal, you can use this direct trick to write it as a fraction:

1. Numerator: Take the entire number after the decimal point and subtract the non-repeating part from it.

   • Entire number after decimal = $13241$

   • Non-repeating part = $13$

   • Numerator = $13241 - 13 = \mathbf{13228}$

2. Denominator: Write as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating digits after the decimal.

   • Number of repeating digits ($2, 4, 1$) = Three 9s ($999$)

   • Number of non-repeating digits ($1, 3$) = Two 0s ($00$)

   • Denominator = $99900$

3. Combine them:
   $\frac{p}{q} = \mathbf{\frac{13228}{99900}}$

   1. Divide by 2 (both numbers are even):
      $13228 \div 2 = 6614$
      $99900 \div 2 = 49950$
      The fraction becomes: $\frac{6614}{49950}$

   2. Divide by 2 again (both numbers are still even):
      $6614 \div 2 = \mathbf{3307}$
      $49950 \div 2 = \mathbf{24975}$

   Correct option:$\frac{3307}{24975}$