At what time between 5 and 6 o'clock are the hands of a clock coinciding?

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Accepted Answer

1. **Understand the formula for the angle between clock hands**:  
   The angle between the hour and minute hands is given by:  
   $$ 	heta = (30H - \frac{11M}{2}) $$,  
   where $$ H $$ is the hour and $$ M $$ is the number of minutes past the hour.

2. **Condition for coincidence**:  
   When the hands coincide, the angle $$ 	heta = 0 $$.

3. **Substitute values**:  
   For 5 o'clock ($$ H = 5 $$):  
   $$
   0 = (30 	imes 5 - \frac{11M}{2})
   $$  
   Simplify:  
   $$
   150 = \frac{11M}{2}
   $$  
   Multiply through by 2:  
   $$
   300 = 11M
   $$  
   Solve for $$ M $$:  
   $$
   M = \frac{300}{11} = 27 \frac{3}{11}.
   $$

4. **Conclusion**:  
   The hands coincide at $$27 \frac{3}{11}$$ minutes past 5.

Thus, the correct answer is **(a) $$27 \frac{3}{11}$$ minutes past 5**.

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