In a 100 m race, A beat B by 10 m and C by 13 m. In a race of 180 m, B will beat C by
(a)5.4 m
(b)4.5 m
(c)5 m
(d)6 m

Hint: To solve the question, we have to analyse the given information to understand the connection between the parameters of A, B, C. To solve further apply distance, speed, time formula to calculate the speeds of A, B, C. Use the obtained speeds of B, C to obtain the distance by which B will defeat C in the race of 180 m.

Complete step-by-step answer:
Let A complete the race of 100 m in t seconds.
Let the speeds of A, B, C be \[{{s}_{a}},{{s}_{b}},{{s}_{c}}\]
We know that the distance travelled by the person is equal to the product of speed of the person and the time taken to travel.
By applying the formula for A, we get
Given that A beats B by 10 m. This implies that at the time A completed the race, B is behind A by 10 m, which concludes that B travelled the distance of 10 m less than 100 m in the time period t seconds.
By applying the above mentioned formula, we get
\[\left( 100-10 \right)={{s}_{b}}t\]
\[90={{s}_{b}}t\]
\[t=\dfrac{90}{{{s}_{b}}}\] ….(1)
Given that A beats C by 13 m. This implies that at the time A completed the race, C is behind A by 13 m, which concludes that C travelled the distance of 13 m less than 100 m in the time period t seconds.
By applying the above mentioned formula, we get
\[\left( 100-13 \right)={{s}_{c}}t\]
\[87={{s}_{c}}t\]
By substituting equation (1) in the above equation, we get
\[87={{s}_{c}}\left( \dfrac{90}{{{s}_{b}}} \right)\]
\[\dfrac{{{s}_{c}}}{{{s}_{b}}}=\dfrac{87}{90}\] …. (2)
In comparison with the speed of B, C, we get that B is faster than C.
Let the time travelled by B to complete 180 m race is \[{{t}^{1}}\]
The time taken to travel distance of 180 m by B with its speed \[{{s}_{b}}\] is equal to
\[180={{s}_{b}}{{t}^{1}}\]
\[{{t}^{1}}=\dfrac{180}{{{s}_{b}}}\]
The distance travelled in the time period by C with its speed \[{{s}_{c}}\] is equal to
\[={{s}_{c}}{{t}^{1}}\]
By substituting \[{{t}^{1}}\] value from equation (3), we get
\[={{s}_{c}}\times \dfrac{180}{{{s}_{b}}}\]
\[=180\times \dfrac{{{s}_{c}}}{{{s}_{b}}}\]
By substituting equation (2) in the above equation, we get
\[\begin{align}
  & =180\times \dfrac{87}{90} \\
 & =2\times 87 \\
 & =174 \\
\end{align}\]
The distance travelled in the time period by C with its speed \[{{s}_{c}}\] is equal to 174 m.
The distance by which B will defeat C = The distance travelled by B - the distance travelled by C in the time period B finished the given race
= 180 - 174
= 6 m
Thus, B will defeat C by 6 m.
Hence, option (d) is the right choice.

Note: The possibility of mistake can be, not analysing the given information that the distance, speed, time formula should be applied to calculate the speeds of A, B, C. The other possibility of mistake can be, making calculation mistake since the procedure of solving involves more concept-oriented calculations. The alternative way of solving is by finding the ratio of distances travelled by A, C and A, B to calculate the ratio of the distance travelled by B, C which when multiplied with the given distance we can calculate the distance travelled by C. Thus, by subtracting the obtained value from the given total distance we will arrive at the answer.