High School

A boy on being asked what \( \frac{16}{17} \) of a fraction was made the mistake of dividing the fraction by \( \frac{6}{7} \), and got an answer, which exceeded the correct answer by \( \frac{33}{340} \). The correct answer is:

(a) \( \frac{68}{85} \)
(b) \( \frac{65}{84} \)
(c) \( \frac{32}{40} \)
(d) None of these

Answer :

To solve the problem, we need to find the correct answer to the fraction problem and determine which given option matches this.

Step-by-step Explanation:


  1. Understanding the problem:


    • The boy made a mistake and instead of finding [tex]\frac{16}{17}[/tex] of a fraction [tex]x[/tex], he divided the fraction by [tex]\frac{6}{7}[/tex].

    • His incorrect answer exceeded the correct answer by [tex]\frac{33}{340}[/tex].



  2. Mathematical representation:


    • The correct operation should be [tex]\frac{16}{17} \times x[/tex].

    • The incorrect operation he performed is [tex]x \div \frac{6}{7} = x \times \frac{7}{6}[/tex].

    • Therefore, the incorrect answer is [tex]x \times \frac{7}{6}[/tex].



  3. Equation setup:


    • According to the problem, [tex]x \times \frac{7}{6} = \frac{16}{17} \times x + \frac{33}{340}[/tex].



  4. Simplifying and solving for [tex]x[/tex]:


    • Equating both sides:
      [tex]x \times \frac{7}{6} = \frac{16}{17} \times x + \frac{33}{340}[/tex]

    • Rearrange the equation:
      [tex]x \times \frac{7}{6} - \frac{16}{17} \times x = \frac{33}{340}[/tex]

    • Find a common factor for the left side, which involves rearranging terms:
      [tex]x \left( \frac{7}{6} - \frac{16}{17} \right) = \frac{33}{340}[/tex]

    • Equivalent fraction subtraction: [tex]\frac{7}{6} - \frac{16}{17}[/tex]

      • Find a common denominator: For [tex]6[/tex] and [tex]17[/tex] is [tex]102[/tex].

      • Convert: [tex]\frac{7}{6} = \frac{119}{102}[/tex] and [tex]\frac{16}{17} = \frac{96}{102}[/tex].

      • Subtract: [tex]\frac{119}{102} - \frac{96}{102} = \frac{23}{102}[/tex].





  5. Solving for [tex]x[/tex]:


    • Replace and solve: [tex]x \times \frac{23}{102} = \frac{33}{340}[/tex]

    • Multiply both sides by [tex]\frac{102}{23}[/tex] to isolate [tex]x[/tex]:
      [tex]x = \frac{33}{340} \times \frac{102}{23}[/tex]

    • Simplify:
      [tex]x = \frac{33 \times 102}{340 \times 23}[/tex]

    • Reduce by finding the GCD (Greatest Common Divisor) and simplifying:
      [tex]x = \frac{11 \times 6}{85 \times 1}[/tex]
      [tex]x = \frac{66}{85}[/tex]



  6. Choosing the correct option:


    • None of the options directly matches our result [tex]\frac{66}{85}[/tex].

    • Therefore, the answer is: (d) None of these.




This solution involves setting up an equation based on the erroneous calculation versus the correct calculation and solving for the unknown fraction [tex]x[/tex].

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