Question

At what time after 7:00 pm will the hands of the clock be opposite to each other?

Answer 1

`7:60/11 \text{pm}`

Explanation:

The minute & hour hands of a clock will be opposite to each other i.e. angle between hands will be `\theta=180^\circ` after `x=7`:00 pm past `y` minutes given as follows

`y=(5x-\theta/6)\times 12/11`

`=(5\times 7-180^\circ/6)\times 12/11`

`=60/11\ \text{minutes}`

Answer 2

Just at 7.00pm the hour hand is at 7 and minute hand is at 12. Each minute division makes `6^@` at the center.

At 7.00pm a reflex angle of `210^@` will be formed between hour hand and minute hand because the hour hand will be at 35 minute divisions ahead of minute hand.
Let x minute after 7.00pm the hands of the clock will be opposite to each other.Then the angle between the hands will be `180^@`

We know hour hand rotates `360^@` in 12hr or `720` minutes

So in x minute the hour hand will increase the reflex angle by `(360x)/720=(x/2)^@`

In this x minute the minute hand will decrease the reflex angle by `(6x)^@` minute.

Hence

`210^@+(x/2)^@-(6x)^@=180^@`

`=>x=(60/11)` minute

So at 7 past `5 5/11`minute the hands will become opposite to each other.