A takes 5 hours more than B to cover a distance of 80 km.  If A doubles his speed, he takes 2\(\frac{1}{2}\) hours more than B to cover 160 km.  To cover a distance of 96 km, how much time will B take traveling at his same speed?

3 hours

\(\frac{80}{A}\) - \(\frac{80}{B}\) = 5

16 (B - A) = AB .... (i)

\(\frac{160}{2A}\) - \(\frac{160}{2B}\) = \(\frac{5}{2}\)

\(\frac{160 [B - 2A]}{2AB}\) = \(\frac{5}{2}\)

⇒ 32 (B - 2A) = AB .... (ii)

From equation (i) and (ii)

32 (B - 2A) = 16 (B - A)

2B - 4A = B - A

⇒ 3A = B

⇒ \(\frac{A}{B}\) = \(\frac{1x}{3x}\)

By equation (i)

\(\frac{80}{x}\) - \(\frac{80}{3x}\) = 5

\(\frac{16 [3 - 1]}{3x}\) = 1

⇒ 3x = 16 × 2

x = \(\frac{32}{3}\)

B will take = \(\frac{96}{\frac{32}{3} × 3}\) = 3 hours