Question

The minute hand of a clock overtakes the hour hand at intervals of 65 minutes. How much in a day does the clock gain 0r lose?
a) Gains $56\dfrac{8}{77}$ minutes
b) Loses $32\dfrac{8}{11}$ minutes
c) Loses $8\dfrac{10}{143}$ minutes
d) Gains $10\dfrac{9}{143}$ minutes
e) Gains $10\dfrac{10}{143}$ minutes

Answer 1

Hint: In order to answer this question, you need to know how the clock works. In a correct or a regular clock, the minute hand takes fifty five minutes spaces per sixty minutes. So, if the clock overtakes the hour hand in suppose N intervals, you can find the gain or loss by using the given formula $\left( \dfrac{720}{11}-N \right)\left( \dfrac{60\times 24}{N} \right)$ minutes. Here, we just have to substitute N as 65 minutes.

Complete step-by-step solution:
In a correct or a regular clock, the minute hand takes fifty five minutes spaces per sixty minutes. Therefore, the minute hand must gain 60 minutes over the hour hand in order to be together again. Therefore, we get that 60 minutes have been gained in $\left( \dfrac{60\times 60}{55} \right)=\dfrac{720}{11}$.
But in the question, the clock only takes 65 minutes. Therefore the gain is given by $\dfrac{720}{11}-65=\dfrac{5}{11}$.
Therefore, the total gain for the day can be found out by multiplying the gain by twenty four and sixty and dividing it by sixty five. Therefore, by doing this, we get:
$\Rightarrow \dfrac{5}{11}\times 24\times \dfrac{60}{65}=\dfrac{1440}{141}=10\dfrac{10}{143}$.
Therefore, we get the answer as $10\dfrac{10}{143}$ minutes, which matches with the E option.
Therefore, the final option is E.

Note: To do this question, you need to know the working of the clock and the basic details as to how many spaces of minutes it takes for the minute hand to complete 60 minutes. Also, you could directly use the formula, where you just have to substitute the N as 65.