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CAT 2023 Slot 1 — QA Question 8

Angle between hands of a clockEasy

The minor angle between the hour hand and minute hand of a clock was observed at 8:48 am. The minimum deviation (in min) after 8:48 am when the angle increases by 50% is?

Answer & solution

  • A

    36/11

  • 24/11

  • C

    2

  • D

    4

Solution

Easy

Use the standard clock formula: the angle between the hands at hh hours mm minutes is ∣30h−112m∣\left|30h-\tfrac{11}{2}m\right|. Compute the angle at 8 ⁣: ⁣488\!:\!48, increase it by 50%50\%, then divide the extra angle by the hands' relative speed of 112\tfrac{11}{2} degrees per minute to get the time.

1

Angle at 8 ⁣: ⁣488\!:\!48:

∣30×8−112×48∣=∣240−264∣=24∘\left|30\times 8-\frac{11}{2}\times 48\right|=\left|240-264\right|=24^\circ
2

A 50%50\% increase means the angle must grow by

0.5×24∘=12∘0.5\times 24^\circ=12^\circ
3

Relative speed of minute vs hour hand:

6∘/min−12∘/min=112 ∘/min6^\circ/\text{min}-\tfrac12{}^\circ/\text{min}=\frac{11}{2}\ {}^\circ/\text{min}

Just after 8 ⁣: ⁣488\!:\!48 the minute hand is closing on, then opening from, the hour hand; the angle next changes at this rate, so the minimum extra time to add 12∘12^\circ is:

t=12 11/2 =2411 mint=\frac{12}{\,11/2\,}=\frac{24}{11}\ \text{min}
t=2411 min(option b)t=\mathbf{\dfrac{24}{11}}\ \text{min}\quad\text{(option b)}

The gap is shrinking at 8 ⁣: ⁣488\!:\!48 (minute hand catching up), so the angle first decreases toward the alignment and only later increases. The quickest way to reach 24+12=36∘24+12=36^\circ is to wait while it changes by 12∘12^\circ at 112∘\tfrac{11}{2}{}^\circ/min once it's increasing, giving 2411\tfrac{24}{11} min — the minimum deviation asked for.

CAT 2023 Slot 1 QA Q8: The minor angle between the hour hand and minute hand of a clock was observed at 8:48 am. The minimum deviatio — Solution | TheCATExam