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

$7 : \frac{60}{11} \setminus \textrm{\pm}$

#### Explanation:

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

$y = \left(5 x - \setminus \frac{\theta}{6}\right) \setminus \times \frac{12}{11}$

$= \left(5 \setminus \times 7 - {180}^{\setminus} \frac{\circ}{6}\right) \setminus \times \frac{12}{11}$

$= \frac{60}{11} \setminus \setminus \textrm{\min u t e s}$

Jul 4, 2018

Just at 7.00pm the hour hand is at 7 and minute hand is at 12. Each minute division makes ${6}^{\circ}$ at the center.

At 7.00pm a reflex angle of ${210}^{\circ}$ 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 ${180}^{\circ}$

We know hour hand rotates ${360}^{\circ}$ in 12hr or $720$ minutes

So in x minute the hour hand will increase the reflex angle by $\frac{360 x}{720} = {\left(\frac{x}{2}\right)}^{\circ}$

In this x minute the minute hand will decrease the reflex angle by ${\left(6 x\right)}^{\circ}$ minute.

Hence

${210}^{\circ} + {\left(\frac{x}{2}\right)}^{\circ} - {\left(6 x\right)}^{\circ} = {180}^{\circ}$

$\implies x = \left(\frac{60}{11}\right)$ minute

So at 7 past $5 \frac{5}{11}$minute the hands will become opposite to each other.