Question

A watch with an hour hand a minute hand shows the time 3:28. how many minutes later will the angle between the 2 hands will be 120 degrees?

Answer 1

At 3 o' clock, the hands are 15 minute spaces apart.
Now, to be at 120$°$ , the minute hand has to gain (15 + 20) minute spaces.

We know that 55 minute spaces are gained in 60 minutes

so, 35 minute spaces are gained in $\left(\frac{60}{55}\times 35\right)\mathrm{minutes}=\frac{420}{11}\mathrm{minutes}$ = $38\frac{2}{11}\mathrm{minutes}$

So, the time at which the angle between hour hand and minute hand is 120$°$ after 3 o' clock is $38\frac{2}{11}\mathrm{minutes}\mathrm{past}3.$

The clock is showing 28 minutes past 3 initially.

So, after $\left(\frac{420}{11}-28\right)\mathrm{minutes}=\frac{112}{11}\mathrm{minutes}=10\frac{2}{11}\mathrm{minutes}$ , the hands of the clock are at 120$°$