At what angle are the hour and minute hands of a clock inclined at 15 minutes past 5?
67½°
Understanding Clock Angles at 5:15
This problem asks us to find the angle between the hour hand and the minute hand of a standard clock at a specific time: 15 minutes past 5, which is 5:15.
To solve this, we need to understand how fast each hand moves around the clock face. The clock face is a circle, covering 360 degrees.
Speed of the Minute Hand
The minute hand completes a full circle (360°) in 60 minutes.
- Speed of minute hand = $\frac{360^\circ}{60 \text{ minutes}} = 6^\circ \text{ per minute}$.
At 15 minutes past 5, the minute hand has moved 15 minutes past the 12 position. The angle of the minute hand from the 12 is:
- Angle of minute hand = $15 \text{ minutes} \times 6^\circ/\text{minute} = 90^\circ$.
The minute hand is pointing exactly at the '3'.
Speed of the Hour Hand
The hour hand completes a full circle (360°) in 12 hours. First, let's find its speed per hour:
- Speed of hour hand per hour = $\frac{360^\circ}{12 \text{ hours}} = 30^\circ \text{ per hour}$.
Since there are 60 minutes in an hour, its speed per minute is:
- Speed of hour hand per minute = $\frac{30^\circ/\text{hour}}{60 \text{ minutes}/\text{hour}} = 0.5^\circ \text{ per minute}$.
At 5:15, the hour hand has moved past the '5' position because 15 minutes have passed since 5:00. At exactly 5:00, the hour hand would be at the '5'. The angle of the '5' mark from the 12 is $5 \times 30^\circ = 150^\circ$.
In the 15 minutes past 5, the hour hand moves an additional angle:
- Additional angle of hour hand = $15 \text{ minutes} \times 0.5^\circ/\text{minute} = 7.5^\circ$.
So, the total angle of the hour hand from the 12 at 5:15 is:
- Total angle of hour hand = Angle at 5:00 + Additional angle = $150^\circ + 7.5^\circ = 157.5^\circ$.
Calculating the Angle Between the Hands
To find the angle between the hour and minute hands, we find the absolute difference between their angles from the 12 mark.
- Angle of minute hand = $90^\circ$
- Angle of hour hand = $157.5^\circ$
The difference is:
- Angle between hands = $|157.5^\circ - 90^\circ| = 67.5^\circ$.
The angle $67.5^\circ$ is the same as $67\frac{1}{2}^\circ$.
Therefore, at 15 minutes past 5 (5:15), the hour and minute hands of a clock are inclined at an angle of 67.5 degrees or 67½ degrees.
| Hand | Speed per hour | Speed per minute |
|---|---|---|
| Minute Hand | $360^\circ$ | $6^\circ$ |
| Hour Hand | $30^\circ$ | $0.5^\circ$ |
Revision Table: Key Concepts for Clock Angle Problems
| Concept | Formula/Value | Explanation |
|---|---|---|
| Total degrees on clock face | $360^\circ$ | A full circle |
| Degrees between numbers (12 to 1, 1 to 2, etc.) | $30^\circ$ | $360^\circ / 12$ hours |
| Minute hand speed | $6^\circ$/minute | $360^\circ / 60$ minutes |
| Hour hand speed | $0.5^\circ$/minute | $30^\circ / 60$ minutes |
| Angle of minute hand from 12 at M minutes past H | $6M^\circ$ | Angle measured clockwise |
| Angle of hour hand from 12 at H hours and M minutes | $(30H + 0.5M)^\circ$ | Angle measured clockwise |
| Angle between hands | $|Hour\ Hand\ Angle - Minute\ Hand\ Angle|$ | Absolute difference |
Additional Information on Clock Angles
Clock angle problems are common in aptitude tests. They require you to calculate the relative positions of the hour and minute hands at a specific time.
The key is to determine the angle of each hand relative to the 12 o'clock position (usually taken as $0^\circ$) and then find the difference. Remember that the hour hand moves continuously, not just on the hour marks. Its position depends on both the hour and the number of minutes past the hour.
Sometimes, the calculated angle might be greater than $180^\circ$. In such cases, the smaller angle between the hands is usually requested, which would be $360^\circ - \text{calculated angle}$. In this specific problem, the calculated angle $67.5^\circ$ is less than $180^\circ$, so it is the direct answer.
Understanding the speeds of the hands ($6^\circ$ per minute for the minute hand and $0.5^\circ$ per minute for the hour hand) is fundamental to solving these problems accurately.
Important Questions from Time, Speed and Distance
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Anuj notices that the reflection of the hands of the wall clock in a mirror is showing the time to be 6 hours 15 minutes. What is the actual time shown by the clock?
The speed of a boat in still water is 5km/h. If it can travel 26 km downstream and 14 km upstream in the same time, then the speed of the stream is:
How many times do the hour hand and the minute hand of a clock coincide in a day?
A person crosses a 700m long bridge in 321 minutes. The speed of the person in km/h is:
