Question

A clock gains 2 minutes every hour. Then the angle traversed by the second hand in one minute is:

• A. $360^\circ$
• B. $370^\circ$
• C. $390^\circ$
• D. $372^\circ$

Hint:

“Gains $g$ minutes per hour” $\Rightarrow$ speed factor $=\dfrac{60+g}{60}$. Multiply the usual angular sweep by this factor.

Answer 1

The Correct Option is D

A gain of $2$ minutes per \emph{real} hour means the clock advances $62$ “clock minutes” in $60$ real minutes. So its rate factor is \[ \text{rate}=\frac{62}{60}=\frac{31}{30}. \] Normally, the second hand sweeps $360^\circ$ in one minute. With the faster rate, in one real minute it sweeps \[ 360^\circ\times\frac{31}{30}=372^\circ. \] Hence, $372^\circ$.

Answer 2

The question asks for the angle swept by a clock's second hand in one real minute, given that the clock gains 2 minutes every hour. A normal second hand sweeps 360 degrees in one minute, but a fast clock's hand moves faster, so we can check which option is consistent with the given gain.

1. \( 360^{\circ} \): This is the sweep on a correctly running clock. Since this clock runs faster than normal, its second hand must sweep more than 360 degrees in one real minute, so this option is too small.
2. \( 370^{\circ} \): A clock gaining 2 minutes in 60 minutes runs at the rate \( \frac{62}{60} \) of normal speed. At this rate, one real minute gives a sweep of \( 360 \times \frac{62}{60}=372 \) degrees, not 370, so this is close but not exact.
3. \( 390^{\circ} \): This would require a speed factor of \( \frac{390}{360}=\frac{13}{12} \), which corresponds to gaining 5 minutes every hour, not 2 minutes, so this option is too large.
4. \( 372^{\circ} \): Since the clock gains 2 minutes every 60 real minutes, it completes 62 clock minutes for every 60 real minutes, a speed factor of \( \frac{62}{60}=\frac{31}{30} \). In one real minute the second hand sweeps \( 360 \times \frac{31}{30}=372 \) degrees, which matches exactly.

The clock's speed factor of \( \frac{31}{30} \) applied to the normal 360 degree sweep gives exactly 372 degrees in one real minute.

Therefore, the correct answer is \( 372^{\circ} \).

Answer 3

A clock that gains 2 minutes every hour sweeps a little more than the usual 360 degrees in a real minute; the extra angle comes from the extra \( \tfrac{2}{60}=\tfrac{1}{30} \) of a minute's worth of movement packed into every real minute. We can check each option by working out what hourly gain it would imply and comparing that to the given 2 minutes per hour.

1. \( 360^{\circ} \): This has no extra angle at all, implying a gain of 0 minutes per hour, which does not match the given 2 minutes.
2. \( 370^{\circ} \): The extra angle here is \( 370-360=10^{\circ} \), which as a fraction of \( 360^{\circ} \) is \( \tfrac{10}{360}=\tfrac{1}{36} \) of a minute's worth of movement; over 60 real minutes this comes to \( 60 \times \tfrac{1}{36} \approx 1.67 \) minutes gained per hour, not 2.
3. \( 390^{\circ} \): The extra angle here is \( 30^{\circ} \), a fraction \( \tfrac{30}{360}=\tfrac{1}{12} \), giving a gain of \( 60 \times \tfrac{1}{12}=5 \) minutes per hour, far more than the given 2 minutes.
4. \( 372^{\circ} \): The extra angle here is \( 12^{\circ} \), a fraction \( \tfrac{12}{360}=\tfrac{1}{30} \), giving a gain of \( 60 \times \tfrac{1}{30}=2 \) minutes per hour, matching the given rate exactly.

Only 372 degrees corresponds to an hourly gain of exactly 2 minutes, as stated in the question.

Therefore, the correct answer is \( 372^{\circ} \).