Question

The hands of a clock coincide after every $66$ minutes of correct time. How much is the clock fast or slow in $24$ hours?

• A. $12\frac{108}{121}$
• B. $11\frac{109}{121}$
• C. $Both\left(A\right)\&\left(B\right)$
• D. None of these

Answer 1

The correct option is B $11\frac{109}{121}$

The angle between $2$ successive numbers in the clock is $\frac{360}{12}={30}^{\circ }$

The angle between successive dots is $\frac{360}{60}=6^{\circ }$

For one rotation of minutes hand hours hand rotates by ${30}^{\circ }$

$⟹$ For increase in $1$ minute there is increase of ${\frac{1}{2}}^{\circ }$ for hours hand

Net decrease For $1$ min is ${\frac{11}{2}}^{\circ }$

For $x$ min the decrease must be ${30}^{\circ }$

$⟹x=\frac{60}{11}$

So the hands to co incide again it takes $60+\frac{60}{11}⟹65\frac{5}{11}$ min

The clock shows $\frac{6}{11}$ min error for $66$ min

The clock shows $x$ min error for $24\times 60$

$⟹x=\frac{\frac{5}{11}\times 24\times 60}{11}=11\frac{109}{121}$