Question

Mira and Amal walk along a circular track, starting from the same point at the same time. If they walk in the same direction, then in 45 minutes, Amal completes exactly 3 more rounds than Mira. If they walk in opposite directions, then they meet for the first time exactly after 3 minutes. The number of rounds Mira walks in one hour is

Answer 1

Let's break the problem down step by step.

Let $r$ be the rate at which Mira walks and $a$ be the rate at which Amal walks. Let the circumference of the circular track be $C$.

1. Walking in the same direction:
   In 46 minutes, the relative distance covered by Amal with respect to Mira (since they're moving in the same direction) is equivalent to 3 rounds. So, $46(a - r) = 3C$
   From this, $a - r = \frac{3C}{46}$

2. Walking in opposite directions:
   When moving in opposite directions, their relative speed gets added.
   So, in 3 minutes, they've covered a distance equivalent to the circumference of the track (because they meet after Amal has walked a full circle more than Mira).
   This means $3(a + r) = C$

From this, $a + r = \frac{C}{3}$ ... (ii)

Now, summing equations (i) and (ii):

$2a = \frac{3C}{46} + \frac{C}{3}$

To get Mira's speed, subtract (i) from (ii):

$2r = \frac{C}{3} - \frac{3C}{46}$

$r = \frac{C}{6} - \frac{3C}{92}$

$r = \frac{11C}{46}$

This means Mira covers a distance equivalent to$\frac{11}{46}$ of the track in one minute.

In 60 minutes (1 hour), she covers $\frac{11 \times 60}{46} = 14.35$ times the circumference of the track.

So, Mira walks 14 rounds in one hour (because we'll only consider the complete rounds).