Question

A can run 1 km in 4 min 54 sec and B in 5 min. How many metres can A give B in a km race so that the race may end in a dead heat?
16 m
18 m
20 m
22 m

Answer 1

Hint:
In this question, both A and B cover the same distance in different time periods. We will find the difference between the time taken by them to know how much time B is taking more than A to cover the same distance. Then, by using distance speed and time formula, we will find the distance covered by B in this extra time taken by him.

Formula Used:
We will use the formula \[d = s \times t\], where \[d\] is the distance, \[s\]is the speed and \[t\] is the time taken by the object to cover the distance.

Complete step by step solution:
Time taken by A to cover 1 km \[ = \] 4 min 54 sec
As \[1{\rm{ min}} = 60{\rm{ sec}}\], we can rewrite above sentence as
Time taken by A to cover 1 km \[ = \left( {4 \times 60} \right)\sec + 54\sec \]
Multiplying the terms, we get
Time taken by A to cover 1 km \[ = \left( {240 + 54} \right){\rm{ sec}}\]
Adding the terms, we get
Time taken by A to cover 1 km \[ = 294{\rm{ sec}}\]
Time taken by B to cover 1 km = 5 min
As \[1{\rm{ min}} = 60{\rm{ sec}}\], we can rewrite above sentence as
Time taken by B to cover 1 km \[ = \left( {5 \times 60} \right){\rm{ sec}}\]
Multiplying the terms, we get
Time taken by B to cover 1 km \[ = 300{\rm{ sec}}\]
Now we will find the difference between the time taken by A and B.
Hence, Difference \[ = 300 - 294 = 6{\rm{ sec}}\]
A covers the same distance in \[6{\rm{ seconds}}\]less than B.
Now we will find speed of A and Busing the formula \[d = s \times t\].
Therefore, Speed of A= \[\dfrac{{1000}}{{294}}{\rm{ m/s}}\]
And, Speed of B= \[\dfrac{{1000}}{{300}}{\rm{ m/s}}\]
Now we will find distance travelled by B in those extra 6 seconds.
Hence, distance travelled by B in those extra 6 seconds \[ = \dfrac{{1000}}{{300}} \times 6{\rm{ m}}\]
Multiplying the terms, we get
Distance travelled by B in those extra 6 seconds \[ = \dfrac{{1000}}{{50}}{\rm{ m}}\]
Dividing the terms, we get
Distance travelled by B in those extra 6 seconds \[ = 20{\rm{ m}}\]
Therefore, A beats B by 20 metre.
Hence, A must give B a start of 20 metre so that the race may end in a dead heat.

Hence, option C is the correct option.

Note:
This is the concept of races and games and we can directly use the following formula to solve the question alternatively.
If A can run \[x\] metres race in \[{T_1}\] sec and B can run the same distance in \[{T_2}\] sec,
Where time taken by A \[ < \] time taken by B
\[{T_1} < {T_2}\]
Then, A beats B by a distance of \[\dfrac{{\left( {{T_2} - {T_1}} \right)}}{{{T_2}}} \times x{\rm{ meters}}\].
Hence, finishing the race in a dead heat.
Substituting \[{T_1} = 294\sec \] and \[{T_2} = 300\sec \] in the above equation, we get
\[ \Rightarrow \] A beats B by a distance \[ = \dfrac{{\left( {300 - 294} \right)}}{{300}} \times 1000\]m
Simplifying the expression, we get
\[ \Rightarrow \] A beats B by a distance \[ = \dfrac{6}{3} \times 10\]m
On simplifying further, we get
\[ \Rightarrow \] A beats B by a distance \[ = 20\]m
Therefore, A beats B by 20 metre.