Question

At what time between $6\text{a.m.}$ and $7\text{a.m.}$ will the minute hand and hour hand of a clock make an angle ${59}^{\circ }$ ?

• A. $6:21\text{a.m.}$
• B. $6:24\text{a.m.}$
• C. $6:23\text{a.m.}$
• D. $6:22\text{a.m.}$

Answer 1

The correct option is D $6:22\text{a.m.}$

At $6:00\text{a.m.},$ the initial angle between minute and hour hand is ${180}^{\circ }$.
We know that the angle made by hour hand in $1$ minute is ${\left(\frac{30}{60}\right)}^{\circ }$ and angle made by minute hand in $1$ minute is ${\left(\frac{360}{60}\right)}^{\circ }$.

Let us assume that after $x$ minutes, the angle is ${59}^{\circ }$.
$\Rightarrow {59}^{\circ }={180}^{\circ }-{\left(\frac{360}{60}\right)}^{\circ }x+{\left(\frac{30}{60}\right)}^{\circ }x$
$\Rightarrow {59}^{\circ }={180}^{\circ }-6^{\circ }x+{0.5}^{\circ }x$
$\Rightarrow x=22\text{min}$
Therefore, required time is $6:22\text{a.m.}$