A has a start of 150 m. So A has to run 1000-150=850 metre while B has to run 1000 metre.
$\text{Speed of A = }4 \dfrac{1}{5}\text{ km/hr }= \dfrac{21}{5}\text{ km/hr }= \dfrac{21}{5} × \dfrac{5}{18} = \dfrac{21}{18} = \dfrac{7}{6}\text{ m/s}$
$\text{Speed of B = 8 km/hour = }8 × \dfrac{5}{18} = \dfrac{20}{9}\text{ m/s}$
$\text{Time taken by B to travel 1000 metre = }\dfrac{\text{distance}}{\text{Speed of B}} = \dfrac{1000}{\left(\dfrac{20}{9}\right)}= \text{ 450 second}$
$\text{Distance travelled by A by this time = time × speed of A = }450 \times \dfrac{7}{6}= 525\text{ metre}$
Hence, A win by $\left(850-525\right)$=325 metre
While A runs 600 m, B runs $\left(600-60\right)$=540 metre
$\text{=>While A runs 400 m, B runs }\dfrac{540}{600} \times 400\text{ = 360 metre}$
While B runs 500 m, C runs $\left(500-50\right)$=450 metre
$\text{=>While B runs 360 m, C runs }\dfrac{450}{500} \times 360\text{ = 324 metre}$
i.e., When A runs 400 metre, B runs 360 metre and C runs 324 metre.
Hence, A beats C by $\left(400-324\right)$=76 metre in a race of 400 m
B has a start of 30 metre
=> A has to run 220 metre and B has to run $\left(220-30\right)$=190 metre
Given that A takes 41 seconds to cover this 220 metre
B takes 44 seconds to cover 220 metre
$\text{=> B takes }\dfrac{44}{220}\text{ seconds to cover 1 metre}$
$\text{=> B takes }\dfrac{44}{220} \times 190 = \dfrac{4}{20} \times 190 \text{ = 38 seconds to cover 190 metre}$
i.e., A beats B by $\left(41-38\right)$= 3 seconds
Let x points make the game
A can give B 20 points, A can give C 32 points
=> When A scores x points, B scores $\left(x-20\right)$ points and C scores $\left(x-32\right)$ pointsB can give C 15 points
=> When B scores x points, C scores $\left(x-15\right)$ points
$\text{=> When B scores 1 point, C scores }\dfrac{(x-15)}{x}\text{ points}$
$\text{=> When B scores $\left(x-20\right)$ point, C scores }\dfrac{(x-20)(x-15)}{x}\text{ points}$
$\text{i.e., } \left(x-32\right) = \dfrac{(x-20)(x-15)}{x}$
$\Rightarrow x(x-32) = (x-20)(x-15)$
$\Rightarrow x^2 - 32x = x^2 - 35x + 300$
$\Rightarrow - 32x = - 35x + 300$
$\Rightarrow 3x = 300$
$\Rightarrow x = 100$
i.e., 100 points make the game
A can give C 18 points in 90
=> While A scores 90 points, C scores $\left(90-18\right)$=72 points
$\text{=> While A scores }\dfrac{90}{72}\text{ points, C scores 1 point.}$
$\text{=> While A scores }\dfrac{90}{72} \times 120 =\dfrac{90}{6} \times 10\text{ = 150 points, C scores 120 point.}$
A can give B 20 points in 60
=> While A scores 60 points, B scores $\left(60-20\right)$=40 points
$\text{=> While A scores 1 point, B scores }\dfrac{40}{60} = \dfrac{2}{3}\text{ points}$
$\text{=> While A scores 150 points, B scores }\dfrac{2}{3} \times 150 = 100\text{ points}$
i.e., while C scores 120 points, B scores 100 points
Hence, in a 120 race, C can give B $\left(120-100\right)$=20 points
In a km race A can beat B by 100 m
=> While A run 1000 m, B run $\left(1000-100\right)$=900 m
In a km race B can beat C by 60 m
=> While B runs 1000 m, C runs $\left(1000-60\right)$=940 m
$\text{ => While B run 1 m, C run }\dfrac{940}{1000}\text{ m}$
$\text{ => While B runs 900 m, C runs }\dfrac{940}{1000} \times 900 = 94 \times 9 = 846\text{ m}$
i.e., while A run 1000 m, C run 846 m
Hence, A can beat C by $\left(1000-846\right)$= 154 m
Let x be the length of the course
A beats B by 15 metres and C by 29 metres
=> When A runs x metre, B runs $\left(x-15\right)$ metre and C runs $\left(x-29\right)$ metreWhen B and C run over the course together, B wins by 15 metres
=> when B runs x metre, C run $\left(x-15\right)$ metre
$\text{=> when B runs 1 metre, C run }\dfrac{x-15}{x}\text{ metre}$
$\text{=> when B runs $\left(x-15\right)$ metre, C run }\dfrac{x-15}{x} \times (x-15) = \dfrac{(x-15)^2}{x}\text{ metre}$
$\text{i.e., }(x-29) = \dfrac{(x-15)^2}{x}$
$\Rightarrow x^2 - 29x = x^2 - 30x + 225$
$\Rightarrow - 29x = - 30x + 225 $
$\Rightarrow x = 225$
i.e., in this race, A will win by $\left(5 min - 4 min 54 sec\right)$ = $\left(300 sec - 294 sec\right)$ = 6 sec
B covers 1000 m in 5 min
=> B covers 1000 m in 300 sec
$\text{=> B covers }\dfrac{1000}{300}= \dfrac{10}{3}\text{m in 1 sec}$
$\text{=> B covers }\dfrac{10}{3} \times 6 = 20\text{ m in 6 sec}$
i.e., A can give B a start of 20 metre so that the race will end in a dead heat.
Speed of A : Speed of B = $\dfrac{11}{8}$: 1 = 11 : 8
Let Speed of A = 11k
and Speed of B = 8k
Let the distance to the goal = x
Time taken by A = time taken by B
=$\dfrac{Distance\: travelled\:by\: A}{Speed\: of\: A}$=$\dfrac{Distance\: travelled\:by\: B}{Speed\: of\: B}$
= $\dfrac{x}{11k}$= $\dfrac{x-90}{8k}$
= $\dfrac{x}{11}$= $\dfrac{x-90}{8}$
8x=11x-990
3x=990
x=330
Distance to the goal = 330 metre
Speed of A = 6 km/hr
=$6\times\dfrac{5}{18}$
=$\dfrac{5}{3}m/s$
Time taken by A = $\dfrac{distance}{speed}$
=$\dfrac{100}{\left(\dfrac{5}{3}\right)}$
=60sec
A gives B a start of 4 m
i.e., B has to cover (100-4)
=96 metre
A beats B by 4 sec
=> Time taken by B to run the race = 60 + 4
= 64 sec
Speed of B =$ \dfrac{Distance}{Time}$
=$\dfrac{96}{64}m/s$
=$\dfrac{96}{64}\times\dfrac{18}{5}$
=5.4 km/hr