At what time between 2 o'clock and 3 o'clock will the hour and minutes hands of a watch be together?

$10\frac{10}{11}$ min past 2

The correct answer is Option (3) → $10\frac{10}{11}$ min past 2

Step-by-Step Calculation:

To find when the hands of a clock coincide (are together), we can use the relative speed formula.

1. Understand the Initial Gap:

At 2 o'clock, the hour hand is at 2 and the minute hand is at 12.

• The distance between them is 10 minute spaces (since each hour mark represents 5 minutes).

2. Relative Speed of the Hands:

• In 60 minutes, the minute hand gains 55 minutes over the hour hand.
• Therefore, to gain 1 minute, the minute hand takes $\frac{60}{55}$ or $\frac{12}{11}$ minutes.

3. Calculate the Time to Close the Gap:

To be "together," the minute hand must gain exactly 10 minutes over the hour hand.

$\text{Time taken} = 10 \times \frac{12}{11} \text{ minutes}$

$\text{Time taken} = \frac{120}{11} \text{ minutes}$

4. Convert to a Mixed Fraction:

Divide 120 by 11:

• $11 \times 10 = 110$
• Remainder = 10
• So, $\frac{120}{11} = 10\frac{10}{11}$ minutes.