A : B = 100 : 90.
A : C = 100 : 72.
B : C =$ \dfrac{B}{A} \times \dfrac{A}{C} $=$ \dfrac{90}{100} \times \dfrac{100}{72} $=$ \dfrac{90}{72} $.
When B runs 90 m, C runs 72 m.
When B runs 100 m, C runs$ \left(\dfrac{72}{90} \times 100\right) $m= 80 m.
$\therefore$ B can give C 20 m.
As speed =$ \left(5 \times\dfrac{5}{18} \right) $m/sec=$ \dfrac{25}{18} $m/sec.
Time taken by A to cover 100 m =$ \left(100 \times\dfrac{18}{25} \right) $sec= 72 sec.
$\therefore$ Time taken by B to cover 92 m = (72 + 8) = 80 sec.
$\therefore$ Bs speed =$ \left(\dfrac{92}{80} \times\dfrac{18}{5} \right) $kmph= 4.14 kmph.
To reach the winning post A will have to cover a distance of $\left(500 - 140\right)$m, i.e., 360 m.
While A covers 3 m, B covers 4 m.
While A covers 360 m, B covers$ \left(\dfrac{4}{3} \times 360\right) $m= 480 m.
Thus, when A reaches the winning post, B covers 480 m and therefore remains 20 m
behind.
$\therefore$ A wins by 20 m.
A : B = 100 : 90.
A : C = 100 : 87.
$ \dfrac{B}{C} $=$ \dfrac{B}{A} \times \dfrac{A}{C} $=$ \dfrac{90}{100} \times \dfrac{100}{87} $=$ \dfrac{30}{29} $.
When B runs 30 m, C runs 29 m.
When B runs 180 m, C runs$ \left(\dfrac{29}{30} \times 180\right) $m= 174 m.
$\therefore$ B beats C by $\left(180 - 174\right)$ m = 6 m.
A : B = 60 : 45.
A : C = 60 : 40.
$\therefore \dfrac{B}{C} $=$ \left(\dfrac{B}{A} \times\dfrac{A}{C} \right) $=$ \left(\dfrac{45}{60} \times\dfrac{60}{40} \right) $=$ \dfrac{45}{40} $=$ \dfrac{90}{80} $= 90 : 80.
$\therefore$ B can give C 10 points in a game of 90.
Speed of A = 2 m/s
$\text{Time taken by A to run 100 m }\dfrac{\text{distance}}{\text{speed}}=\dfrac{100}{2}\text{ = 50 seconds}$
A gives B a start of 4 metres and still A beats him by 10 seconds
=> B runs $\left(100-4\right)$=96 m in $\left(50+10\right)$=60 seconds
$\text{Speed of B = }\dfrac{\text{distance}}{\text{time}}=\dfrac{96}{60}\text{ = 1.6 m/s}$
P run 1 km in 3 minutes
Q run 1 km in 4 minutes 10 secs
$\text{=> Q runs 1 km in }\dfrac{25}{6}\text{ minutes}$
$\text{=> Q runs }\left(1 \times \dfrac{6}{25} \times 3\right)= \dfrac{18}{25} = 0.72\text{ km in 3 minutes}$
Hence, in a 1 km race, P can give Q $\left(1 - 0.72\right)$=0.28 km = 280 metre
While A runs 100 metre, B runs $\left(100-10\right)$=90 metre
While B runs 100 metre, C runs $\left(100-5\right)$=95 metre
$\text{=> While B runs 90 metre, C run }\dfrac{95}{100} \times 90 = \dfrac{95 \times 9}{10}\text{ = 85.5 metre}$
ie, when A run 100 metre, B run 90 metre and C run 85.5 metre
Hence, A beat C by $\left(100-85.5\right)$= 14.5 metre
While X runs 1000 metre, Y runs $\left(1000-52\right)$=948 metre and Z runs $\left(1000-83\right)$=917 metre
i.e., when Y runs 948 metre, Z runs 917 metre
$\text{=> When Y runs 1000 metre, Z runs }\dfrac{917}{948} \times 1000\text{ = 967.30 metre}$
i.e., Y can give Z $\left(1000-967.30\right)$ = 32.7 metre
This means, B takes 4 seconds to run 40 metres
$\text{=> B takes }\dfrac{4}{40} = \dfrac{1}{10}\text{ seconds to run 1 metre}$
$\text{=> B takes }\dfrac{1}{10} \times 1000 = 100 \text{ seconds to run 1000 metre}$