Question

On a given day, how many times will the second-hand and the minute-hand of a clock cross each other during the clock time 12{:05{:}00 hours to 12{:}55{:}00 hours?}

• A. 51
• B. 49
• C. 50
• D. 55

Hint:

For clock-hand crossing counts over a specific interval, write the exact congruence with an integer $k$ instead of just dividing by the period; the initial offset (here $30^\circ$) can add an extra crossing at the start.

Answer 1

The Correct Option is C

Step 1: Set angles and speeds.
Second hand speed = $6^\circ/\text{s}$, minute hand speed = $0.1^\circ/\text{s}$.
At $12{:}05{:}00$, minute hand is at $5 \times 6^\circ = 30^\circ$, second hand at $0^\circ$.
Let $t$ seconds after $12{:}05{:}00$. Crossing occurs when $6t \equiv 30+0.1t \pmod{360}$, i.e. $5.9t = 30 + 360k,\; k \ge 0$.
Hence $t_k = (30+360k)/5.9$.

Step 2: Count all crossings in the window.
We need $0 < t_k \le 3000$ (since $12{:}55{:}00$ is $3000$ s later).
Inequality: $(30+360k)/5.9 \le 3000 \;\Rightarrow\; 30+360k \le 17700 \;\Rightarrow\; k \le 49$.
So $k = 0,1,2,\ldots,49$ — a total of 50 crossings.

Therefore, $\boxed{50}$.