In a 3000m race, if vehicle A gives vehicle B a start of 400m, then vehicle A wins the race by 10s. If vehicle A gives vehicle B a start of 750m, then the race ends in a dead heat. How long does vehicle A take to run 3000m?

Let t
A
​
be the time taken by vehicle A to run the full race of 3000m, and v
B
​
be the speed of vehicle B.

Case 2: A gives B a start of 750m and the race ends in a dead heat.
This means B needs to cover (3000−750)=2250m in the same time t
A
​
that A takes to cover 3000m.

Speed of B (v
B
​
)=
t
A
​

2250
​

Case 1: A gives B a start of 400m and wins by 10s.
This means B needs to cover (3000−400)=2600m, and takes 10s more than A's time.

Time taken by B to run 2600m=t
A
​
+10
v
B
​

2600
​
=t
A
​
+10
Substitute v
B
​
from Case 2:

(
t
A
​

2250
​
)
2600
​
=t
A
​
+10
2250
2600
​
t
A
​
=t
A
​
+10
225
260
​
t
A
​
=t
A
​
+10
45
52
​
t
A
​
−t
A
​
=10
45
7
​
t
A
​
=10
t
A
​
=
7
450
​
≈64.285s≈64.3s