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).

Answer 2

Let 'a' represent the amount of time Amal needs to finish a round and'm' represent the amount of time Mira needs to finish a round.
Amal finished 45 rounds in 45 minutes, or \(\frac{45}{a}.\) 
Number of rounds completed by Mira in 45 minutes = \(\frac{45}{m}.\) 

We are aware that Amal finished three rounds more than Mira in forty-five minutes. 
Consequently, \(\frac{45}{a} = \frac{45}{m} + 3 \quad \text{and} \quad \frac{45}{a} - \frac{45}{m}\)\(= 3 \)
\(\frac{1}{a} - \frac{1}{m} = \frac{1}{15} \quad \text{(1)}\)

Amal finished three rounds in three minutes, or \(\frac{3}{a}\). 
Mira finished a total of \(\frac{3}{m}\) rounds in 3 minutes. 

We are aware that Amal and Mira can do one round together in three minutes. 
It follows that \(\frac{3}{a} + \frac{3}{m} = 1\)
\(\frac{1}{a} + \frac{1}{m} = \frac{1}{3} \quad \text{(2)}\)
\((2) - (1) \)
=\(\frac{1}{a} + \frac{1}{m} - \frac{1}{a} + \frac{1}{m}\)

=\(\frac{1}{3} - \frac{1}{15}\)

\(\frac{2}{m} = \frac{4}{15}\)

\(\frac{1}{m} = \frac{2}{15}\)

\(m = \frac{15}{2}\)
Mira thus requires \(\frac{15}{2}\) minutes to finish one round. 

It takes her 15 minutes to do each round and 60 minutes to finish 8.
In an hour, Meera completes 8 rounds.