__Category__: Algebra"Published in Newark, California, USA"

A bus bound for Baguio left Manila at exactly 5:00 am traveling at a uniform speed. Thirty minutes later, a car left Manila in pursuit of the bus. Traveling also at a uniform rate, the car overtook the bus 100 miles from the city. Had it increased its speed by 10 miles per hour, it would have overtaken the bus only 50 miles from Manila. Find the rate at which the bus was traveling.

__Solution__:

Well, this is a complicated problem solving about the rate, distance, and time problem. We have to analyze the given word problem as follows

Let x = be the speed of a bus

y = be the speed of a car

t = time

Let's draw a simple figure to illustrate the problem

Photo by Math Principles in Everyday Life |

As you noticed that we subtract 30 minutes or ½ hour for the travel time of a car because there's a 30 minute gap between a bus and a car. Since a bus left Manila first, then we have to subtract ½ hour for the travel time of a car.

Distance = speed x time

From the word statement "the car overtook the bus 100 miles from the city", the working equation for the said statement is

Distance of a Bus = Distance of a Car

The total time traveled by a car at uniform speed is

If the speed of a car is increased by 10 miles per hour, then the total time will be

Excess Time = Total Time at Uniform Speed - Total Time at Increased Speed

From the word statement "Had it increased its speed by 10 miles per hour, it would have overtaken the bus only 50 miles from Manila", then the working equation will be

Distance = Rate x Time

If you choose a positive sign,

Total time traveled by a car

Therefore, the speed of a bus is

If you choose a negative sign,

Total time traveled by a car

Therefore, the speed of a bus is