Solution:
Let the length of the tunnel be x m
Time = $$\frac{{{\text{Length of train}} + {\text{Length of tunnel}}}}{{{\text{Speed}}}}$$
60 = $$\frac{{350 + {\text{x}}}}{{10}}$$
$$ {{\text{Speed}} = \frac{{36 \times 5}}{{18}} = 10\,\,{\text{m/sec}}} $$
x = 250 meters
Therefore length of tunnel is 250 meters

Solution:
Downstream speed, = $$\frac{9}{2}$$ = 4.5 kmph Upstream speed,= $$\frac{9}{6}$$ = 1.5 kmph Thus, speed of boat in still water, = $$\frac{{4.5 + 1.5}}{2}$$   = 3 km/h Speed of current = $$\frac{{4.5 - 1.5}}{2}$$   = 1.5 km/h

Solution:
$$\eqalign{ {\text{Distance}} = \frac{{{S_1} \times {S_2}}}{{{S_1} - {S_2}}} \times {\text{difference of time}} \cr {\text{240}} = \frac{{S\left( {S - 8} \right)}}{{S - S + 8}} \times 1 \cr 240 \times 8 = S \times \left( {S - 8} \right) \cr 48 \times 40 = S \times \left( {S - 8} \right) \cr S = 48{\text{ km/hr}}{\text{.}} \cr {\text{ = }}\frac{{480}}{{48}} = 10{\text{ hrs}} \cr} $$

Solution:
(6 - x) = (8 - 1.5x) Or, x = 4 cm. So, It will take 4 hours to burn it in such a way that they will remain equal in lengths.

Solution:
When A runs 1000m, B runs 900m. Hence, when A runs 500m, B runs 450m. Again, when B runs 400m, C runs 360m. And, when B runs 450m, C runs = 360 × $$\frac{{450}}{{400}}$$ = 405m. Required distance = 500 - 405 = 95 meter. That means when A runs 500 meter then B can run 450m then C runs 405m.

Solution:
Let A's speed = x km/hr
Then, B's speed = (7 - x) km/hr
So,
$$\eqalign{ \Leftrightarrow \frac{{24}}{x} + \frac{{24}}{{\left( {7 - x} \right)}} = 14 \cr \Leftrightarrow 24\left( {7 - x} \right) + 24x = 14x\left( {7 - x} \right) \cr \Leftrightarrow 14{x^2} - 98x + 168 = 0 \cr \Leftrightarrow {x^2} - 7x + 12 = 0 \cr \Leftrightarrow \left( {x - 3} \right)\left( {x - 4} \right) = 0 \cr \Leftrightarrow x = 3{\text{ or }}x = 4 \cr} $$
Since A is faster than B,
So, A's speed = 4 km/hr
And B's speed = 3 km/hr

Solution:
Let the usual speed be x km/hr
Then,
$$\eqalign{ \Leftrightarrow \frac{{75}}{x} - \frac{{75}}{{x + 10}} = \frac{{15}}{{60}} \cr \Leftrightarrow x\left( {x + 10} \right) = 3000 \cr \Leftrightarrow {x^2} + 10x - 3000 = 0 \cr \Leftrightarrow \left( {x + 60} \right)\left( {x - 50} \right) = 0 \cr \Leftrightarrow x = 50 \cr} $$
∴ Required time :
$$\eqalign{ = \left( {\frac{{300}}{{60}}} \right){\text{hrs}} \cr = 5{\text{ hrs}} \cr} $$