Question

At what time after 12:00 o’clock midnight will the minute

hand and the hour hand of a clock be on a straight line for

the first time?

Answer 1

Since the Complete Angle is $360^{\circ}$; therefore, the minute hand travels $360^{\circ}$in one hour.

The clock is divided into $12$ parts (hour); therefore, the hour hand travels $\frac{360^{\circ}}{12}=30^{\circ}$ in one hour.

Let $x$ be the position of the minute hand on the clock (in minutes)

Since one hour has $60$ minutes; therefore, one hour can be expressed as $\frac{x}{60}$.

So to find the position of the minute hand on the clock substitute into the relation:

Min hand in degrees - Hour hand in degrees = $180^{\circ}$

$\frac{x}{60}\cdot 360-\frac{x}{60}\cdot 30=180^{\circ}$

Simplify each fraction:

$6x-\frac{1}{2}x=180$

Subtract the terms:

$5.5x=180$

Divide both sides by $5.5$ :

$x\approx 32.73$

So, at $12:32:73$ both pm, the hands of a clock will be in a straight line.