Understanding the Time Between 7 and 8 O'clock When the Hands of a Watch Point in Opposite Directions

The intriguing question often arises: at what precise time between 7 and 8 o'clock will the hands of a watch point in opposite directions? To solve this, we'll dissect the mechanics of a clock's hands and perform a detailed calculation.

The Hour and Minute Hands' Movements

To solve the problem, we need to understand the movement of the hour and minute hands:

Hour Hand

The hour hand moves at a rate of 30 degrees per hour, given that 360 degrees (a full circle) divided by 12 hours equals 30 degrees per hour. Additionally, the hour hand moves 0.5 degrees per minute since 30 degrees divided by 60 minutes equals 0.5 degrees per minute.

Minute Hand

The minute hand moves at a rate of 6 degrees per minute, derived from 360 degrees divided by 60 minutes.

Initial Positions and Calculation

At 7:00, the initial positions of the hands are:

The hour hand is at 7 times 30 degrees, which equals 210 degrees. The minute hand is at 0 degrees (at the 12 o'clock position).

Determining the Equation for Opposite Directions

The hands are opposite each other when they are 180 degrees apart. Let ( x ) represent the number of minutes past 7:00. The positions of the hands at 7:x can be described as follows:

The hour hand is at 210 0.5x degrees. The minute hand is at 6x degrees.

Setting Up the Equation for Opposite Directions

We can set up the equation by considering the distance between the hands:

For opposite directions:

( 210 0.5x - 6x 180 ) OR ( 6x - (210 0.5x) 180 )

Solving the First Case

First, we solve the equation ( 210 0.5x - 6x 180 ):

( 210 - 5.5x 180 )
( -5.5x -30 )
( x frac{30}{5.5} frac{300}{55} frac{60}{11} approx 5.45 ) minutes

Solving the Second Case

Next, we solve the equation ( 6x - (210 0.5x) 180 ):

( 6x - 210 - 0.5x 180 )
( 5.5x - 210 180 )
( 5.5x 390 )
( x frac{390}{5.5} frac{3900}{55} 70.91 ) minutes

Since 70.91 minutes exceeds the hour, we discard this case as it's not feasible.

Conclusion

The hands of the clock will point in opposite directions at approximately 7:05.45 minutes. This can be expressed in a more standard time format as 7 minutes and 27 seconds past 7 o'clock. For those interested, we can use a more intuitive approach to verify this solution.

Another Solution Method

Another way to look at it is that at 7 o'clock, the minute hand is ahead of the hour hand by 25 minutes. When the hands are in opposite directions, the minute hand is ahead by 5 minutes. Since the minute hand advances 55 minutes to cover a 60-minute gap, it will be ahead by 5 minutes in ( frac{5 times 12}{11} 5 frac{5}{11} ) minutes.

Therefore, the hands will be in opposite directions at 5 (frac{5}{11}) minutes past 7 o'clock.

This solution is equally valid and can be used to double-check the previous result.

Key Takeaways:
1. The hour hand moves 0.5 degrees per minute.
2. The minute hand moves 6 degrees per minute.
3. The hands are 180 degrees apart in opposite directions.