The minor angle between the hours hand and minutes hand of a clock was observed at 8:48 am. The minimum duration, in minutes, after $8.48$ am when this angle increases by $50 \%$ is

$\frac{24}{11}$

$\frac{36}{11}$

✅ Correct Option: 1

We need to find when the angle between clock hands at 8:48 AM increases by 50%.

To find the angle between hour and minute hands, we use:

$\text{Angle} = \left|M × \tfrac{11}{2} - 30H\right|$

Where M = minutes past the hour and H = hour (in 12-hour format).

The minute hand moves 6° per minute, the hour hand moves 0.5° per minute, giving a relative speed of 5.5° = $\tfrac{11}{2}$ ° per minute. The hour hand starts 30H degrees ahead from 12 o'clock.

At 8:48 AM with H = 8 hours and M = 48 minutes:

$\text{Angle} = 48 × \tfrac{11}{2} - 30 × 8$

$= 48 × 5.5 - 240$

$= 264 - 240 = 24°$

We need to find when this angle increases by 50%:

$\text{New Angle} = 24° + (50\% \text{ of } 24°)$

$= 24° + 12° = 36°$

We need to find the time M when the angle equals 36°:

$36 = M × \tfrac{11}{2} - 30 × 8$

$36 = M × 5.5 - 240$

$276 = M × 5.5$

$M = \tfrac{276}{5.5} = \tfrac{552}{11} = 50\tfrac{2}{11} \text{ minutes}$

Time difference = New time - Original time

$= 50\tfrac{2}{11} - 48$

$= \tfrac{552}{11} - \tfrac{528}{11}$

$= \tfrac{24}{11} \text{ minutes}$

Converting to mixed number: $\tfrac{24}{11} = 2\tfrac{2}{11}$ minutes

The minimum duration after 8:48 AM when the angle increases by 50% is $\tfrac{24}{11}$ minutes or approximately 2.18 minutes.

At 8:48 AM the angle is 24°. After $\tfrac{24}{11}$ minutes, at approximately 8:50 AM, the angle becomes 36°, which is indeed 50% more than 24°.

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The minor angle between the hours hand and minutes hand of a clock was observed at 8:48 am. The minimum duration, in minutes, after 8.488.488.48 am when this angle increases by 50%50 \%50% is

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The minor angle between the hours hand and minutes hand of a clock was observed at 8:48 am. The minimum duration, in minutes, after $8.48$ am when this angle increases by $50 \%$ is