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

Correct Answer: 8

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).