At what time between 7 and 8 will the hands of a clock be in the same straight…

On a clock face divided into 60 minute-spaces, the minute hand travels 60 spaces every hour while the hour hand travels only 5 spaces, so the minute hand gains 55 minute-spaces on the hour hand every 60 minutes. The two hands lie in the same straight line but point in opposite directions (a 180-degree angle) exactly when the gap between them equals half the dial, that is, 30 minute-spaces.

1. At 7:00 sharp, the hour hand rests at the 35-minute-space mark (7 × 5 = 35) and the minute hand rests at the 0-mark, so the short-way gap between them is 60 − 35 = 25 minute-spaces.

2. For the hands to become exactly opposite, this gap must grow from 25 spaces to 30 spaces.

3. So the minute hand only needs to gain 30 − 25 = 5 minute-spaces on the hour hand.

4. Since the minute hand gains 55 minute-spaces every 60 minutes, the time needed to gain 5 spaces is (60/55) × 5 = 300/55 = 60/11 minutes.

5. 60/11 minutes equals 5 and 5/11 minutes, so the hands become exactly opposite at 5 (5/11) minutes past 7.

Independent check using hand angles: at 5 (5/11) minutes past 7, the hour hand has moved 210 + 0.5 × 60/11 = 212 and 8/11 degrees from the 12 mark, and the minute hand has moved 6 × 60/11 = 32 and 8/11 degrees from the 12 mark; the difference between them is exactly 180 degrees, confirming the hands are in a straight line.

Therefore, the hands of the clock are in the same straight line but not together at 5 (5/11) minutes past 7.