distance

[demoskopico]

A man cycling along the road noticed that every 12 minutes a bus overtakes him while every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at constant speed, what is the time interval between consecutive buses?

5 minutes
6 minutes
8 minutes
9 minutes
10 minutes


What is the fastest way to solve problems like this?

[JohnB]

Let's call the distance between each bus 1 km. (It doesn't matter what the distance is, as long as it's constant.) Then we can set up the following two equations based on the problem, where B stands for the rate of the bus and C stands for the rate of the bicyclist:

a) B - C = 1/12 (In other words, the bus travels at an apparent rate of 1km/12 min when C's speed is subtracted from it.)

b) B + C = 1/4 (The bus travels at an apparent rate of 1km/4 min when C's speed is added to it.)

(Why subtraction in the first and addition in the second? Because the bus seems to go slower when it travels in the same direction as C, and seems to go faster traveling in the opposite direction.)

Now, solve this system of equations for B:

c) 12 B - 12 C = 1 (multiplying (a) through by 12)
d) 12 B + 12 C = 3 (multiplying (b) through by 12)
e) 24 B = 4 (adding (c) and (d))
f) B = 1/6.

So the bus's actual rate is 1km every 6 minutes. The answer is 6.