How to Calculate the Degrees Between the Hour and Minute Hands at 3:15

One of the classic interview questions posed by J.P. Morgan is to determine the angle between the hour and minute hands of a clock at 3:15. While it might seem complex at first, understanding the basic principles can lead to a straightforward solution. This article will walk you through the calculation step-by-step and explain the logic behind it.

Understanding the Movement of the Clock Hands

Clocks are divided into 12 hours, with each hour marking a 30-degree sector since (360 div 12 30) degrees. The minute hand moves in smaller increments, covering 360 degrees in 60 minutes, or 6 degrees per minute. The hour hand, on the other hand, moves more slowly, covering 30 degrees per hour, or 0.5 degrees per minute.

Calculating the Angle at 3:15

Let's break down the calculation for the angle at 3:15 step-by-step:

1. Calculating the Position of the Minute Hand

The minute hand points at 15 minutes, which means it has moved 90 degrees from the 12 o'clock position. This can be calculated as follows:

(text{Minute angle} frac{360}{60} times 15 90 text{ degrees})

2. Calculating the Position of the Hour Hand

First, we calculate where the hour hand is at 3:00:

(text{Hour angle at 3:00} 3 times 30 90 text{ degrees})

Since the hour hand continues to move as the minutes pass, we need to account for the additional movement of the hour hand during the 15 minutes. This calculation is as follows:

(text{Additional hour angle} frac{30}{60} times 15 7.5 text{ degrees})

Therefore, at 3:15, the hour hand has moved an additional 7.5 degrees beyond the 90-degree mark. So, the total hour angle at 3:15 is:

(text{Total hour angle} 90 7.5 97.5 text{ degrees})

3. Calculating the Angle Between the Two Hands

The angle between the hour and minute hands is the difference between the total hour angle and the minute angle:

(text{Angle} 97.5 - 90 7.5 text{ degrees})

Alternative Calculation Method

Another method can be used to verify the same result. We can calculate the position of the hour hand in minutes from the 12 o'clock position:

195 minutes (since 3 hours and 15 minutes is 195 minutes) can be used to determine the angle of the hour hand.

(text{Angle for the hour hand} frac{195 times 30}{720} 97.5 text{ degrees})

For the minute hand, we have:

(text{Angle for the minute hand} frac{15 times 6}{1} 90 text{ degrees})

Therefore, the angle between the hour and minute hands is:

(97.5 - 90 7.5 text{ degrees})

Conclusion

At 3:15, the angle between the hour and minute hands is 7.5 degrees. This calculation is a good example of how basic mathematical principles can be applied to solve real-world problems. Familiarizing yourself with such concepts can be particularly useful in interviews and problem-solving scenarios.