Question

Manish has a stick of length 1 metre. He marks the stick at thirds (to divide the stick into 3 equal pieces) with blue pen and at fifths (to divide the stick into 5 equal pieces) with red pen. He then cuts the stick at all the blue and red markings and hence the stick is divided into 7 pieces. What is the length (in metre) of the smallest piece?

• A. \(\frac{1}{5}\)
• B. \(\frac{1}{15}\)
• C. \(\frac{2}{15}\)
• D. \(\frac{2}{5}\)

Answer 1

The Correct Option is B

Step 1: Identify all the cut points on the 1-metre stick 

The stick is marked at:

• Thirds: \( \frac{1}{3}, \frac{2}{3} \).
• Fifths: \( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \).

The combined set of marking points is:

\( 0, \frac{1}{5}, \frac{1}{3}, \frac{2}{5}, \frac{3}{5}, \frac{2}{3}, \frac{4}{5}, 1 \).

Step 2: Convert all values to a common denominator (15)

• \( \frac{1}{5} = \frac{3}{15} \)
• \( \frac{1}{3} = \frac{5}{15} \)
• \( \frac{2}{5} = \frac{6}{15} \)
• \( \frac{3}{5} = \frac{9}{15} \)
• \( \frac{2}{3} = \frac{10}{15} \)
• \( \frac{4}{5} = \frac{12}{15} \)

Step 3: Find the smallest segment

The stick is divided at points:

\( 0, \frac{3}{15}, \frac{5}{15}, \frac{6}{15}, \frac{9}{15}, \frac{10}{15}, \frac{12}{15}, 1 \).

The smallest segment length is the difference between consecutive values:

• \( \frac{5}{15} - \frac{3}{15} = \frac{2}{15} \)
• \( \frac{6}{15} - \frac{5}{15} = \frac{1}{15} \) (smallest)
• \( \frac{9}{15} - \frac{6}{15} = \frac{3}{15} \)
• ...

Final Conclusion:

The smallest piece has a length of \( \frac{1}{15} \), so the correct answer is (B) \( \frac{1}{15} \).