Question

Two persons A and B walk around a circle whose diameter is 1.4km. A walks at a speed of 165 meters per minute while B walks at a speed of 110 meters per minute. If they both start at the same time from the same point and walk in the same direction, at what interval of time would they both be at the same starting point again?
$
  {\text{A}}{\text{. }}1h \\
  {\text{B}}{\text{. }}1\dfrac{1}{3}h \\
  {\text{C}}{\text{. }}1\dfrac{2}{3}h \\
  {\text{D}}{\text{. }}1\dfrac{1}{2}h \\
 $

Answer 1

Hint: Here we go through by the help of the properties of the circle we have to apply the formula of circumference of the circle to find out the distance travelled and then apply the formula of time taken by the given speed.

Complete step by step solution:
Here in the question it is given that two persons A and B walk around a circle whose diameter is 1.4km. A walks at a speed of 165 meters per minute while B walks at a speed of 110 meters per minute.
The diameter of the circle=1.4cm
$\therefore r = \dfrac{{1.4}}{2}km$
So we know that the formula of circumference of the circle i.e. $2\pi r$.
$\therefore $ Circumference of the circle=$2\pi r = 2 \times \dfrac{{22}}{7} \times \dfrac{{1.4}}{2}km = 4.4km$
In the question it is given that a speed is 165m/min.
Now we have to first convert the data in the same unit so we change the kilometer to meter.
I.e. 4.4km=$4.4 \times 1000 = 4400m$ as we know 1km=1000m.
$\therefore $Time taken by A to travel 4.4 km (4400 m) =$\dfrac{{{\text{distance}}}}{{{\text{speed}}}} = \dfrac{{4400}}{{165}}\min = \dfrac{{80}}{3}\min $
And the speed of B is also given as 110m/min.
So similar,
Time taken by B's to travel 4.4 km (4400 m) = $\dfrac{{{\text{distance}}}}{{{\text{speed}}}} = \dfrac{{4400}}{{110}}\min = 40\min $
And now for finding the interval of time that they both are at the same starting point again we have to find out the lowest common multiple of the both times of A and B.
I.e. required interval of time = LCM of $\left( {\dfrac{{80}}{3},40} \right)$
And we know that the process of finding the LCM of a fraction number that we have to find out the LCM of the numerator and we have to divide it by the HCF of the denominator.
i.e. $\dfrac{{LCM(80,40)}}{{HCF(3,1)}} = \dfrac{{80}}{1}$
$\therefore $ Required interval of time= 80min=$\dfrac{{80}}{{60}}hrs = 1\dfrac{1}{3}hrs$
Hence option B is the correct answer.

Note: Whenever we face such a type of question the key concept for solving the question is to first find out the individual time of each person to complete the one rotation and then find out their LCM to find out the time at which they are at the starting point again.