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 \( a \) be the number of minutes required by Amal to complete one round, and \( m \) be the number of minutes required by Mira to complete one round.

In 45 minutes:
Number of rounds completed by Amal = \( \dfrac{45}{a} \)
Number of rounds completed by Mira = \( \dfrac{45}{m} \)

It is given that Amal completed 3 more rounds than Mira in 45 minutes:
\[ \dfrac{45}{a} = \dfrac{45}{m} + 3 \] Subtracting: \[ \dfrac{45}{a} - \dfrac{45}{m} = 3 \Rightarrow \dfrac{1}{a} - \dfrac{1}{m} = \dfrac{1}{15} \tag{1} \]

Also, it is given that in 3 minutes, together they complete 1 round:
\[ \dfrac{3}{a} + \dfrac{3}{m} = 1 \Rightarrow \dfrac{1}{a} + \dfrac{1}{m} = \dfrac{1}{3} \tag{2} \]

Now solving equations (1) and (2) together:
From (1) and (2): \[ \left( \dfrac{1}{a} + \dfrac{1}{m} \right) - \left( \dfrac{1}{a} - \dfrac{1}{m} \right) = \dfrac{1}{3} - \dfrac{1}{15} \Rightarrow \dfrac{2}{m} = \dfrac{4}{15} \Rightarrow \dfrac{1}{m} = \dfrac{2}{15} \Rightarrow m = \dfrac{15}{2} = 7.5 \]

Therefore, Mira takes 7.5 minutes to complete one round.
To find how many rounds Mira covers in 1 hour (60 minutes): \[ \text{Number of rounds} = \dfrac{60}{7.5} = 8 \]

So, Mira covers 8 rounds in 1 hour.

Answer 2

Let a be the time (in minutes) Amal takes to complete one round, and let m be the time Mira takes to complete one round.

In 45 minutes, Amal completes \(\frac{45}{a}\) rounds, and Mira completes \(\frac{45}{m}\) rounds. 

According to the question, Amal completes 3 rounds more than Mira in 45 minutes.

Therefore, we have the equation:
\(\frac{45}{a} = \frac{45}{m} + 3\)
Subtracting both sides:
\(\frac{45}{a} - \frac{45}{m} = 3\)
Divide the whole equation by 45:
\(\frac{1}{a} - \frac{1}{m} = \frac{1}{15} \quad \text{(1)}\)

Now, in 3 minutes, Amal completes \(\frac{3}{a}\) rounds and Mira completes \(\frac{3}{m}\) rounds.

We are also told that together, Amal and Mira complete 1 round in 3 minutes:
\(\frac{3}{a} + \frac{3}{m} = 1\)
Divide the equation by 3:
\(\frac{1}{a} + \frac{1}{m} = \frac{1}{3} \quad \text{(2)}\)

Adding equations (1) and (2):
\(\left( \frac{1}{a} + \frac{1}{m} \right) - \left( \frac{1}{a} - \frac{1}{m} \right) = \frac{1}{3} - \frac{1}{15}\)
Simplifying:
\(\frac{2}{m} = \frac{4}{15} \Rightarrow \frac{1}{m} = \frac{2}{15}\)
Therefore:
\(m = \frac{15}{2}\)

So, Mira takes 7.5 minutes (or \(\frac{15}{2}\) minutes) to complete one round.

In 60 minutes, she will complete:
\(\frac{60}{7.5} = 8\) rounds.

Conclusion: Mira completes 8 rounds in one hour.