Considering the distance travelled by Mira in one minute $= \M$,
The distance traveled by Amal in one minute $= \A$.
Given if they walk in the opposite direction it takes 3 minutes for both of them to meet. Hence $3×(\A+\M) = \C$
...(1)
$\C$ is the circumference of the circle.
Similarly, it is mentioned that if both of them walk in the same direction Amal completes 3 more rounds than Mira :
Hence $45×(\A-\M) = 3\C$
...(2)
Multiplying $(1)×15$ we have :
$45\A + 45\M = 15\C$
$45\A - 45\M = 3\C$
Adding the two we have A = ${18\C}/90$
Subtracting the two M = ${12\C}/90$
Since Mira travels ${12\C}/90$ in one minute, in one hour she travels : ${12\C}/90 . 60 =80C$
Hence a total of 8 rounds.
Alternatively,
Let the length of track be $\L$
and velocity of Mira be $a$ and Amal be $b$
Now when they meet after 45 minutes Amal completes 3 more rounds than Mira.
So we can say they met for the $3^{rd}$ time moving in the same direction
so we can say they met for the first time after 15 minutes
So we know -
$\Time \to \meet = {\Relative \distance} /{\Relative \velocity}$
so we get -$15/60 = \L/{\a.\b}$
...(1)
Now When they move in opposite direction
They meet after 3 minutes
So we get -
$3/60 = \L/{\a +\b }$
...(2)
Dividing (1) and (2)
We get -
${a +b}/ {a -b} = 5 $
or $4a =6b $
or$ a = {3b}/2$
Now substituting in (1)
we get -
$\L/b × 2 = 15/60 $
so, $\L/b = 1/8$ So we can say 1 round is covered in $1/8$ hours
so in 1hour total rounds covered $= 8$.