A : B = 100 : 90.

A : C = 100 : 72.

B : C = \(\frac{B}{A}\times\frac{A}{C}=\frac{10}{100}\times\frac{100}{72}=\frac{90}{72}.\)

When B runs 90 m, C runs 72 m. \(\left(\frac{90}{72}\times100\right)m = 80m.\)

When B runs 100 m, C runs

So, B can give C 20 m.

A's speed = \(\left(5\times\frac{5}{18}\right)m/sec = \frac{25}{18}m/sec\)

Time taken by A to cover 100 m = \(\left(100\times\frac{18}{25}\right)m/sec = 72 sec.\)

so, Time taken by B to cover 92 m = (72 + 8) = 80 sec.

B's speed = \(\left(\frac{92}{80}\times\frac{18}{5}\right)kmph =4.14 kmph\)

To reach the winning post A will have to cover a distance of (500 - 140)m, i.e., 360 m.

While A covers 3 m, B covers 4 m.

While A covers 360 m, B covers \(\left(\frac{3}{4}\times360\right)m =480m.\)

Thus, when A reaches the winning post, B covers 480 m and therefore remains 20 m behind.

So,  A wins by 20 m.

A : B = 100 : 90.

A : C = 100 : 87.

\(\frac{B}{C}=\frac{B}{A}\times\frac{A}{C}=\frac{90}{100}\times\frac{100}{87}=\frac{30}{29}.\)

When B runs 30 m, C runs 29 m.

When B runs 180 m, C runs \(\left(\frac{29}{30}\times180\right)m. = 174m.\)

So, B beats C by (180 - 174) m = 6 m.

A : B = 60 : 45.

A : C = 60 : 40.

\(\frac{B}{C}=\left(\frac{B}{A}\times\frac{A}{C}\right)=\left(\frac{45}{60}\times\frac{60}{40}\right)=\frac{45}{40}=\frac{90}{80}= 90:80.\)

So, B can give C 10 points in a game of 90.

A : B = 200 : 169.

A : C = 200 : 182.

\(\frac{B}{C}=\left(\frac{C}{A}\times\frac{A}{B}\right)=\left(\frac{182}{200}\times\frac{200}{169}\right)=182:169.\)

When C covers 182 m, B covers 169 m.

When C covers 350 m, B covers \(\left(\frac{169}{350}\times350\right)m=325m
\)

Therefore, C beats B by (350 - 325) m = 25 m.

Distance covered by B in 9 sec. = \(\left(\frac{100}{45}\times9\right)m=20m\)

So, A beats B by 20 metres.

A : B = 100 : 80.

A : C = 100 : 72.

So, \(\frac{B}{C}=\left(\frac{B}{A}\times\frac{A}{C}\right)=\left(\frac{80}{100}\times\frac{100}{72}\right)=\frac{10}{9}=\frac{100}{90}=100:90.\)

So, B can give C 10 points.

B runs 35 m in 7 sec.

So,  \(\left(\frac{7}{35}\times200\right)=40sec.\)

B's time over the course = 40 sec.

So, A's time over the course (40 - 7) sec = 33 sec.

When B runs 25 m, A runs \(\frac{45}{2}m.\)

When B runs 1000 m, A runs \(\left(\frac{45}{2}\times\frac{1}{25}\times1000\right)m=900m.\)

So, B beats A by 100 m.