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HomeClass 10mathematicsChapter 1Exercise 1.1Q7
| Exercise 1.1

Question 7

There is a circular path around a sports field. Sonia takes 1818 minutes to drive one round of the field, while Ravi takes 1212 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

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SOLUTION

UNDERSTAND THE QUESTION

The problem asks for the smallest amount of time after which both Sonia and Ravi, traveling around the same circular path at constant speeds, will be back together at the starting point. This requires finding the least common multiple (LCM) of their individual lap times (18 min and 12 min).

STEP 1 — RECORD EACH PERSON'S LAP TIME

Sonia completes one lap in 1818 minutes, so her lap time is 1818 minutes per lap. Ravi completes one lap in 1212 minutes, so his lap time is 1212 minutes per lap.

STEP 2 — PRIME FACTORISE THE LAP TIMES

18=2×3218 = 2 \times 3^{2}
12=22×312 = 2^{2} \times 3

STEP 3 — DETERMINE THE LCM USING THE HIGHEST POWERS OF EACH PRIME

Take the highest power of each prime that appears in the factorizations:

  • For prime 22, the highest power is 222^{2} (from 1212).
  • For prime 33, the highest power is 323^{2} (from 1818). Thus
LCM=22×32=4×9=36.\text{LCM}=2^{2} \times 3^{2}=4 \times 9 = 36.

STEP 4 — INTERPRET THE LCM AS THE MEETING TIME

The LCM of 1818 and 1212 minutes is the smallest time that is an integer multiple of both lap times. Therefore after 3636 minutes each has completed an integer number of laps (Sonia 3618=2\dfrac{36}{18}=2 laps, Ravi 3612=3\dfrac{36}{12}=3 laps) and they meet again at the starting point.

ANSWER

36 minutes
36\text{ minutes}

COMMON MISTAKES

Students often add the two times (18 + 12) instead of finding the LCM, which gives 30 minutes—incorrect because 30 is not a multiple of 18. Another pitfall is forgetting to use the highest powers of each prime when computing the LCM, leading to a smaller value such as 12 minutes.

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