Mira and Amal walk along a circular track, starting from the same point at the sometime. 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

Mira and Amal are walking on a circular track. We need to find how many rounds Mira completes in one hour using two key pieces of information:

Same direction: Amal completes 3 more rounds than Mira in 45 minutes

Opposite direction: They meet for the first time after 3 minutes

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Let's define:

Track length = $x$ units

Mira's speed = $m$ units per minute

Amal's speed = $a$ units per minute

Since we're dealing with circular motion, it's easier to work with speeds and track length rather than trying to count rounds directly.

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When both walk in the same direction, Amal gains on Mira because he's faster.

The difference in distance covered = difference in speeds × time

In 45 minutes:

Distance covered by Amal = $45a$

Distance covered by Mira = $45m$

Difference in distance = $45a - 45m = 45(a - m)$

Since Amal completes exactly 3 more rounds than Mira:

$45(a - m) = 3x$

$a - m = \frac{3x}{45} = \frac{x}{15}$ ... (Equation 1)

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When they walk in opposite directions, they approach each other.

They meet when their combined distance equals one complete track length.

In 3 minutes:

Distance covered by Amal = $3a$

Distance covered by Mira = $3m$

Combined distance = $3a + 3m = 3(a + m)$

Since they meet for the first time after covering one complete track:

$3(a + m) = x$

$a + m = \frac{x}{3}$ ... (Equation 2)

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From Equation 1: $a - m = \frac{x}{15}$

From Equation 2: $a + m = \frac{x}{3}$

Adding both equations:

$(a - m) + (a + m) = \frac{x}{15} + \frac{x}{3}$

$2a = \frac{x}{15} + \frac{5x}{15} = \frac{6x}{15} = \frac{2x}{5}$

Therefore: $a = \frac{x}{5}$

Subtracting Equation 1 from Equation 2:

$(a + m) - (a - m) = \frac{x}{3} - \frac{x}{15}$

$2m = \frac{5x}{15} - \frac{x}{15} = \frac{4x}{15}$

Therefore: $m = \frac{2x}{15}$

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Now we know Mira's speed is $\frac{2x}{15}$ units per minute.

Time for Mira to complete one round:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{x}{\frac{2x}{15}} = \frac{x \times 15}{2x} = 7.5 \text{ minutes}$

Number of rounds in one hour (60 minutes):

$\text{Rounds} = \frac{60}{7.5} = 8$

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Mira walks 8 rounds in one hour.