In a 200-m race, if A gives B a head start of 35m, then A wins the race by 20sec. Alternatively, if A gives B a head start of 55m, the race ends in a dead heat. How long does A take to run 200m?

Let the speed of A be v
A
​
and the speed of B be v
B
​
.
Let the time taken by A to cover 200m be t
A
​
.

Case 2: A gives B a head start of 55m and the race ends in a dead heat.
This means that in the time t
A
​
that A takes to cover 200m, B covers (200−55)=145m.
Therefore, the time taken by B to run 145m is t
A
​
.

Speed of B (v
B
​
)=
t
A
​

145
​

Case 1: A gives B a head start of 35m, and A wins by 20sec.
This means B runs (200−35)=165m, and takes 20sec more than A's total time.

Time taken by B to cover 165m=t
A
​
+20
v
B
​

165
​
=t
A
​
+20
Substitute the value of v
B
​
from Case 2 into this equation:

(
t
A
​

145
​
)
165
​
=t
A
​
+20
145
165
​
t
A
​
=t
A
​
+20
29
33
​
t
A
​
−t
A
​
=20
29
4
​
t
A
​
=20
t
A
​
=
4
20×29
​
=5×29=145sec
Thus, A takes 145sec to run 200m.