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Question

If \(x = \frac{1}{2 + \frac{3}{4 + \frac{5}{6 + \frac{7}{8 + \frac{9}{10}}}}}\), then which one of the following is correct?

This question was previously asked in
CDS 2 2024 Maths Question Paper (01-Sep-2024)
The correct answer is
$0 < x < 0.5$

Evaluating Continued Fraction \(x\)

The question asks us to find the range in which the value of \(x\) lies, where \(x\) is defined by the continued fraction:

\( x = \frac{1}{2 + \frac{3}{4 + \frac{5}{6 + \frac{7}{8 + \frac{9}{10}}}}} \)

To determine the range, we evaluate the continued fraction step by step, starting from the innermost part and working outward. This method helps us approximate the value of \(x\) accurately.

Step-by-Step Evaluation

  1. Innermost part: \( 8 + \frac{9}{10} = 8.9 \)
  2. Next level: \( 6 + \frac{7}{8.9} \approx 6 + 0.7865 = 6.7865 \)
  3. Next level: \( 4 + \frac{5}{6.7865} \approx 4 + 0.7367 = 4.7367 \)
  4. Next level: \( 2 + \frac{3}{4.7367} \approx 2 + 0.6334 = 2.6334 \)
  5. Final value: \( x = \frac{1}{2.6334} \approx 0.3797 \)

Determining the Range

The approximate value of \(x\) is \(0.3797\). Comparing with the given options:

  • \(0 < x < 0.5\) ✓ (Correct)
  • \(x = 0.5\) ✗
  • \(0.5 < x < 1\) ✗
  • \(x > 1\) ✗

Bounding Method (Rigorous Approach)

Upper bound:

  • \(8 + \frac{9}{10} > 8 \Rightarrow \frac{7}{8 + \frac{9}{10}} < \frac{7}{8}\)
  • \(6 + \frac{7}{\cdots} < 6.875\)
  • \(\frac{5}{\cdots} < \frac{5}{6}\)
  • \(4 + \cdots < 4.833\)
  • \(\frac{3}{\cdots} < \frac{3}{4}\)
  • \(2 + \cdots < 2.75\)
  • \( x > \frac{1}{2.75} = \frac{4}{11} \approx 0.3636 \)

Lower bound:

  • \(8 + \frac{9}{10} < 9 \Rightarrow \frac{7}{\cdots} > \frac{7}{9}\)
  • \(6 + \cdots > \frac{61}{9}\)
  • \(\frac{5}{\cdots} > \frac{45}{61}\)
  • \(4 + \cdots > \frac{289}{61}\)
  • \(\frac{3}{\cdots} > \frac{183}{289}\)
  • \(2 + \cdots > \frac{761}{289}\)
  • \( x < \frac{289}{761} \approx 0.3797 \)

Thus, we obtain:

\( 0.3636 < x < 0.3797 \)

Conclusion

The value of \(x\) lies in the range:

\( 0 < x < 0.5 \)

Hence, the correct option is \(0 < x < 0.5\).

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Important Questions from Fractions

  1. Simplify the expression 441 ÷  \(\left[270 \div \frac{3}{7}+\left(17\div \frac{1}{3}\right)-\left(8\frac{1}{2}-\frac{5}{2}\right)\right]\)

  2. Number 0.232323 can be written in rational form as:

  3. Solve: \(\frac{1}{2}\)  [{-2(2 + 3)*20}/2]

  4. Match the following.

    Column I

    Column II

    a.

    Equivalent fraction of \(\frac{7}{12}\)  is  

    i.

    Proper fraction

    b.

    Equivalent fraction of  \(\frac{9}{15}\)  is

    ii.

    Improper fraction

    c.

    \(\frac{7}{11}\)  is

    iii.

    \(\frac{21}{36}\)

    d.

    \(\frac{19}{5}\)  is

    iv.

    \(\frac{3}{5}\)

  5. Among the fractions \(\frac{7}{12}\) , \(\frac{8}{15}\) ,  \(\frac{17}{24}\)  and \(\frac{13}{18}\) , which is the smallest?

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