In a 1500 m race, if vehicle P gives vehicle Q a start of 200m, vehicle P wins the race by 8 sec. Alternatively, if vehicle P gives vehicle Q a start of 400 m, the race ends in a dead heat. How long does vehicle P take to run 1500 m ?

44 sec

Let’s assume speed of P = x and speed of Q = y

Then, in a 1500 m race distance run by P= 1500 m and distance run by Q after getting head start of 200m = 1300m

Time taken by P = \(\frac{1500}{x}\)

Time taken by Q = \(\frac{1300}{y}\)

Since Q lacks behind by 8 sec therefore Time taken by Q is more as compare to that of P by 8 sec,

    \(\frac{1300}{y}\) -  \(\frac{1500}{x}\)  = 8    ----(1)

When Q gets head start of 400m they both finish the race at same time,

Time taken by Q = \(\frac{1500 - 400}{y}\)= \(\frac{1100}{y}\)

Time taken by P = \(\frac{1500}{x}\)

Both finish at same time,

 \(\frac{1100}{y}\) = \(\frac{1500}{x}\)

x   = \(\frac{15}{11}\) × y       ----(2)

Putting (2) in (1) we get,

\(\frac{200}{y}\) = 8

y = 25 m /sec

And x = \(\frac{375}{11}\) m sec

Time taken by P to run 1500 m = \(\frac{1500 × 11 }{375}\)

= 44 sec