Question

Ali bicycles to a place 20km away. John's route to the same place is 4km longer but he rides 2km/h faster than Ali and gets their 5mins later. What are their speeds?

Answer 1

Assume Ali's speed is $v$ , then John's speed is $v + 2$ , km/h.

Time spent: by Ali is $\cfrac{20}v$ , by John is $\cfrac{20 + 4}{v + 2}$ ., h.

Given that John spent 5 minutes (1/12 hour) more, we can make the equation

$\qquad \cfrac{20}{v} + \cfrac{1} {12} = \cfrac{20+4}{v + 2}$

To get rid of fractions we multiply the left and right sides of the equation by $12v(v+2)$

$\qquad 20 * 12 (v+2) + 1*v(v+2) = 24*12v$

$\qquad v^2 - 46v + 480 = 0$

$\qquad (v -16)(v - 30) = 0$

$\qquad \underline{v_1 = 16}, \space \underline{v_2 = 30}$

The problem has two solutions: 

1) Ali's speed is 16 km/h, John's speed is 18 km/h.

2) Ali's speed is 30, John's speed is 32.