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Hare and Tortoise Race Separations

XAT 2020 · Quantitative Ability & Data Interpretation · Hard · Quantitative Aptitude

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A hare and a tortoise run between points O and P located exactly 6 km from each other on a straight line. They start together at O, go straight to P and then return to O along the same line. They run at constant speeds of 12 km/hr and 1 km/hr respectively. Since the tortoise is slower than the hare, the hare shuttles between O and P until the tortoise goes once to P and returns to O. During the run, how many times are the hare and the tortoise separated by an exact distance of 1 km from each other?

  1. A.

    24

  2. B.

    42

  3. C.

    22

  4. D.

    48

  5. E.

    40

Answer

E

Explanation

Total distance for tortoise = 6+6=126 + 6 = 12 km. At speed 1 km/hr, total time = 12 hours. Hare speed = 12 km/hr, so total distance = 12×12=14412 \times 12 = 144 km = 12 complete rounds of O→P→OO \to P \to O.

In each round of the hare (12 rounds total):

  • Round 1: 3 occurrences of 1 km separation.
  • Rounds 2 to 5: 4 occurrences each   ⟹  4×4=16\implies 4 \times 4 = 16.
  • Round 6: 2 occurrences. First half = 3+16+1=203 + 16 + 1 = 20 occurrences. By symmetry, the second half (rounds 7 to 12) has another 20 occurrences. Total occurrences = 20×2=4020 \times 2 = 40.

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