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Tuesday, February 5, 2013

Rate, Distance, Time - Problem

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



If you drive faster, you will arrive to your destination earlier. In this case, you save time for commuting or driving if you drive faster and you will have an excess time as well. Excess time is calculated as follows

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