Question

A and B travel around in a circular path at uniform speeds in the opposite direction starting from the diagrammatically opposite point at the same time. They meet each other first after B has traveled 100 meters and meet again 60 meters before A complete one round. What is the circumference of the circular path?
A. $240\,{\text{m}}$
B. $360\,{\text{m}}$
C. $480\,{\text{m}}$
D. $300\,{\text{m}}$

Answer 1

Determine the distances travelled by A and B in both the cases. Take the ratio of these distances and equate them as their speed is uniform.

Complete step by step answer: Here, we have to calculate the circumference of the circular path travelled by A and B.

Let the half of the circumference of the circular path is $x$.

Initially, A and B start from the opposite points of any diameter of the circular track. They meet each other when B has travelled a distance of $100\,{\text{m}}$ on the half circumference.

Hence, the distance ${x_{B1}}$ travelled by B on the half circumference is $100\,{\text{m}}$ and distance travelled ${x_{A1}}$ by A is$\left( {x - 100} \right)\,{\text{m}}$.
${x_{B1}} = 100\,{\text{m}}$
${x_{A1}} = \left( {x - 100} \right)\,{\text{m}}$

Take the ratio of ${x_{A1}}$ and ${x_{B1}}$.
$\dfrac{{{x_{A1}}}}{{{x_{B1}}}} = \dfrac{{\left( {x - 100} \right)\,{\text{m}}}}{{100\,{\text{m}}}}$

A and B once again meet each when A is $60\,{\text{m}}$ before to complete one round.

Hence, the distance ${x_{A2}}$ travelled by A on the half circumference is $\left( {2x - 60} \right)\,{\text{m}}$ and distance travelled ${x_{B2}}$ by B is$\left( {x + 60} \right)\,{\text{m}}$.
${x_{A2}} = \left( {2x - 60} \right)\,{\text{m}}$
${x_{B2}} = \left( {x + 60} \right)\,{\text{m}}$

Take the ratio of ${x_{A2}}$ and ${x_{B2}}$.
$\dfrac{{{x_{A2}}}}{{{x_{B2}}}} = \dfrac{{\left( {2x - 60} \right)\,{\text{m}}}}{{\left( {x + 60} \right)\,{\text{m}}}}$

Since A and B travel with uniform speed on the circular track, the ratios $\dfrac{{{x_{A1}}}}{{{x_{B1}}}}$ and $\dfrac{{{x_{A2}}}}{{{x_{B2}}}}$ of the distances covered by A and B are equal.
$\dfrac{{{x_{A1}}}}{{{x_{B1}}}} = \dfrac{{{x_{A2}}}}{{{x_{B2}}}}$

Substitute $\dfrac{{\left( {x - 100} \right)\,{\text{m}}}}{{100\,{\text{m}}}}$ for $\dfrac{{{x_{A1}}}}{{{x_{B1}}}}$ and $\dfrac{{\left( {2x - 60} \right)\,{\text{m}}}}{{\left( {x + 60} \right)\,{\text{m}}}}$ for $\dfrac{{{x_{A2}}}}{{{x_{B2}}}}$ in the above equation.
$\dfrac{{\left( {x - 100} \right)\,{\text{m}}}}{{100\,{\text{m}}}} = \dfrac{{\left( {2x - 60} \right)\,{\text{m}}}}{{\left( {x + 60} \right)\,{\text{m}}}}$

Solve the above equation for $x$.
$\left( {x - 100} \right)\,\left( {x + 60} \right) = \left( {2x - 60} \right)100$
$\Rightarrow {x^2} + 60x - 100x - 6000 = 200x - 6000$
$\Rightarrow x\left( {x - 240} \right) = 0$
$\Rightarrow x = 0$ or $x = 240$

Since, the value of the half circumference cannot be zero.
$x = 240\,{\text{m}}$

Therefore, the half circumference of the circular path is $240\,{\text{m}}$.

Since the circumference is twice the half circumference, $2x = 2\left( {240\,{\text{m}}} \right) = 480\,{\text{m}}$.

Hence, the circumference of the circular path is $480\,{\text{m}}$.

Hence, the correct option is D.

Note: The value zero obtained when solved the quadratic equation should be neglected as the circumference cannot be zero for the present question.