Question

At what time are the hands of the clock together between 5 and 6?

Answer 1

Hint: Take 5 O’clock in an initial situation. At this point, the hands of the clock are 25 minutes apart from each other. Also, we need to use the information that 1 hour is equal to 60 minutes. In a clock, 60 minutes means 1 circular round of minute hand. The point is when the minute hand moves 60 steps at the same time hour hand moves only 5 steps (from the perspective of 1 minute as 1 step)

Complete step-by-step answer:
The Idea to solve this question is to observe the obvious thing that is 1 hour = 60 minutes
It means as the minute hand moves 60 steps, the hour hand moves 5 steps but slower than minute hand. Which can also be seen as
60 minute-steps = 5 hour-steps
So, 1 minute-steps = ${\displaystyle \frac{5}{60}}={\displaystyle \frac{1}{12}}$ hour-steps

Let us assume it takes y minute-steps.
So, this y minute step should be equal to steps from 12 to 5 which is 25 steps plus the steps moved by hour hand. Which is $m\times {\displaystyle \frac{1}{12}}$. This can also be written as
$\begin{array}{c}m=25+m\times {\displaystyle \frac{1}{12}}\\ \Rightarrow m=25+{\displaystyle \frac{m}{12}}\end{array}$
We can solve this linear equation with unknown m to get our required answer.
Multiplying the whole equation by 12 and taking m-involving terms together.
$\begin{array}{c}12m=12\times 25+m\\ \Rightarrow 12m-m=300\\ \Rightarrow 11m=300\\ \Rightarrow m={\displaystyle \frac{300}{11}}\end{array}$
On dividing 300 by 11 we’ll the minutes at which two hands of the watch coincide.
$m={\displaystyle \frac{300}{11}}=27.\overline{27}$
Hence, two hands of the watch coincide between 5 to 6 at 27.3 minutes approximately.

Note: The hack in this question is to observe that as the minute hand moves and travels the distance till 5 the hour hand also moves. So, to get the answer we have to find a time where both meets. One can also solve this by the time and distance concept.