Multiple choice

Rajesh went out between $5$ and $6$ and returned between $6$ and $7$. He found that the hands of the clock had exactly changed places. When did Rajesh go out?

  1. $31 \displaystyle \frac{4}{13}$ min. past $5$
  2. $33 \displaystyle \frac{4}{13}$ min. past $5$
  3. $32 \displaystyle \frac{4}{13}$ min. past $5$
  4. $32 \displaystyle \frac{6}{13}$ min. past $5$
  5. $32 \displaystyle \frac{5}{13}$ min. past $5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

When clock hands change places, the time T (in minutes past 5) satisfies the condition that the hour hand is at 5 + T/60 and the minute hand is at T. After swapping, the new positions are T/12 (for the hour hand) and 5*5 + T/12 (for the minute hand). Solving the system leads to T = 300/13 + 300/13, resulting in 32 4/13 minutes past 5.

AI explanation

Using the swapped hands formula, the sum of the fractions of the hour intervals covered by the two hands must equal the total time difference, meaning the departure minute is given by M equals 60 times the hour difference plus the initial hour gap divided by 13. Between 5 and 6, substituting the values gives 60 times 1 plus 4 divided by 13, which equals 64 divided by 13. Converting this improper fraction to a mixed number yields 4 over 13, meaning he went out at 32 and 4 over 13 minutes past 5.

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