Question

What will be the measurement of the angle made by the hour and minute hands of a clock when the time is ‘quarter past 3’?

• A. $6 \frac{1}{2}$
• B. 10
• C. $8 \frac{1}{2}$
• D. $7 \frac{1}{2}$

Answer 1

The Correct Option is D

To determine the angle between the hour and minute hands when the time is "quarter past 3", follow these steps: 

1. Minute hand position: When the time is 3:15, the minute hand is at the 15-minute mark. Each minute represents 6 degrees (since 360 degrees/60 minutes = 6 degrees per minute). Thus, the position is:
15 minutes × 6 degrees/minute = 90 degrees.

2. Hour hand position: The hour hand moves 30 degrees per hour (since 360 degrees/12 hours = 30 degrees per hour). At exactly 3:00, the hour hand is at 90 degrees (3 hours × 30 degrees/hour). Since 15 minutes is 1/4 of an hour, the hour hand moves a bit further:
(1/4) × 30 degrees = 7.5 degrees.
Thus, at 3:15, the hour hand is at:
90 degrees + 7.5 degrees = 97.5 degrees.

3. Calculating the angle between the hands: The angle between the hour and minute hands is the absolute difference between their positions:
|97.5 degrees - 90 degrees| = 7.5 degrees.

The angle made by the hour and minute hands at 3:15 is therefore $7 \frac{1}{2}$ degrees.

Answer 2

Understanding 'Quarter Past 3'

'Quarter past 3' means 3:15 on the clock.

Calculating Minute Hand Position

At 15 minutes, the minute hand points to the 3 on the clock.

Each number on the clock represents \( 30^\circ \) (since \( \frac{360^\circ}{12} = 30^\circ \)).

Minute hand angle: \( 3 \times 30^\circ = 90^\circ \) from 12 o'clock position.

Calculating Hour Hand Position

At 3:00, the hour hand points exactly at the 3 (\( 90^\circ \)).

In 15 minutes, the hour hand moves \( \frac{1}{4} \) of the way to the next number (4).

Hour hand movement: \( \frac{30^\circ}{4} = 7.5^\circ \).

Total hour hand angle: \( 90^\circ + 7.5^\circ = 97.5^\circ \).

Calculating the Angle Between Hands

Difference between hands: \( |97.5^\circ - 90^\circ| = 7.5^\circ \).

We take the smaller angle between the two possible angles (the other being \( 360^\circ - 7.5^\circ = 352.5^\circ \)).

Verification

Using the clock angle formula:

\[ \theta = |30H - 5.5M| = |30 \times 3 - 5.5 \times 15| = |90 - 82.5| = 7.5^\circ \]

The correct answer is option (3) \( 7 \frac{1^\circ}{2} \).