Question 13

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

** Correct Answer:** 8

Solution

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) = 3C. (2)

Multiplying (1)*15 we have :

45A + 45M = 15C.

45A - 45M = 3C.

Adding the two we have A = $$\frac{18C}{90}$$

Subtracting the two M = $$\frac{12C}{90}$$

Since Mira travels $$\frac{12C}{90}$$ in one minute, in one hour she travels :$$\frac{12C}{90}\cdot60\ =\ 8C$$

Hence a total of 8 rounds.

Alternatively,Â

Let the length of track be LÂ

and velocity of Mira be a and Amal be bÂ

Now when they meet after 45 minutes Amal completes 3 more rounds than Mira

so we can say they met for the 3rd 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 $$\frac{15}{60}=\frac{L}{a-b}$$ Â Â (1)

Now When they move in opposite direction

They meet after 3 minutes

so we get $$\frac{3}{60}=\frac{L}{a+b}$$ Â Â (2)

Dividing (1) and (2)

we get $$\frac{\left(a+b\right)}{\left(a-b\right)}=5$$

or 4a =6b

or a = 3b/2

Now substituting in (1)

we get :

$$\frac{L}{b}\times\ 2=\ \frac{15}{60}$$

so $$\frac{L}{b}\ =\frac{1}{8}$$

So we can say 1 round is covered in $$\frac{1}{8}$$ hours

so in 1-hour total rounds covered = 8.

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