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
- Innermost part: \( 8 + \frac{9}{10} = 8.9 \)
- Next level: \( 6 + \frac{7}{8.9} \approx 6 + 0.7865 = 6.7865 \)
- Next level: \( 4 + \frac{5}{6.7865} \approx 4 + 0.7367 = 4.7367 \)
- Next level: \( 2 + \frac{3}{4.7367} \approx 2 + 0.6334 = 2.6334 \)
- 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 \(0 < x < 0.5\).
Important Questions from Fractions
If three-fifths of a number is 54, what is two-ninth of it?
Sunila had \(9\frac{1}{4}\) kg of flour to make bread with. If the recipe says that she needs \(1\frac{1}{8}\) kg to make one loaf of bread, how many loafs can she make? Estimate to the nearest whole number.
The value of \(\frac{{11}}{5} - \left( {\frac{2}{3}of\frac{3}{5} - \frac{1}{5}} \right) + \left( {\frac{6}{5} \div \frac{4}{5}} \right)\) is:
In his career, a tennis player won 5 matches lost 12 matches and had 3 matches as draw. The fraction of the match he lost in his career is:
Simplify:
\(62\div 5 - \left(\frac{8}{9}\times \frac{18}{7}\right)\times \frac{7}{5}+\frac{5}{4}\times \frac{16}{25}\)
