At what time between 7 o clock and 8 o clock will the hands of a clock be in the same straight line but not together

Answer

To find the time between 7 o'clock and 8 o'clock when the hands of the clock are in the same straight line but not together, we can follow these steps:

### Step 1: Understand the positions of the clock hands
At 7:00, the hour hand is at 210 degrees (since each hour represents 30 degrees, and 7 hours × 30 degrees = 210 degrees). The minute hand is at 0 degrees.

### Step 2: Determine the relative speeds of the clock hands
The minute hand moves at 6 degrees per minute (360 degrees in 60 minutes), while the hour hand moves at 0.5 degrees per minute (30 degrees in 60 minutes).

### Step 3: Set up the equation for the hands being in a straight line
For the hands to be in a straight line but not together, they can either be 180 degrees apart. We can express this mathematically as:
- Position of minute hand = Position of hour hand + 180 degrees
- Position of minute hand = 0 + 6m (where m is the number of minutes after 7:00)
- Position of hour hand = 210 + 0.5m

Setting up the equation:
\[ 6m = 210 + 0.5m + 180 \]

### Step 4: Simplify the equation
Combine like terms:
\[ 6m - 0.5m = 210 + 180 \]
\[ 5.5m = 390 \]

### Step 5: Solve for m
\[ m = \frac{390}{5.5} \]
\[ m = 70.91 \text{ minutes} \]

However, since we are only looking for the time between 7:00 and 8:00, we need to find the equivalent time in minutes:
\[ m = 70.91 - 60 = 10.91 \text{ minutes} \]

### Step 6: Convert minutes to time format
10.91 minutes is approximately 10 minutes and 55 seconds. Therefore, the time when the hands are in the same straight line but not together is around 7:10:55.

### Final Answer
The hands of the clock will be in the same straight line but not together at approximately **7:10:55**.

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