Intro to Distance, Rate & Time Problems

This past weekend I visited my parents. I was on a stretch of highway about five miles from the edge of the small town where they live when I was passed by four cars in the space of a few minutes — which would be understandable had I been going under the speed limit or even at the speed limit, but I was going 5 mph over the speed limit.

I found their urgency mathematically interesting.

I began to wonder how much time the cars saved by passing me? Should I speed up? Would it make much of a difference?

The Four Cars Passing Problem

• The speed limit was 55 mph.
• I was traveling at 60 mph.
• The cars that passed me traveled ~65 mph.
• There was 5 miles of highway before entering city limits.

We’ll be using an important formula: distance = rate • time.

The distance-rate-time formula

Since we are interested in solving for time, isolate the t-variable beforehand.

5 Miles at 55 MPH

For the first example, I’ll write out the long-hand math so you can see how the labels interact.

Begin by filling in distance and rate in the above formula.

Invert and multiply. Notice the “miles” labels cancel out leaving “hours”.

Now convert 0.0909 hours into minutes.

And lastly change .45 minutes into seconds.

So it would take 5 minutes and 27 seconds to travel 5 miles at 55 mph.

5 Miles at 60 MPH

Now we’ll calculate the time for a car traveling 60 mph for 5 miles.

Convert to minutes.

5 Miles at 65 MPH

Convert into minutes and seconds.

At 65 mph, one would arrive at the city limit in 4 minutes and 37 seconds.

The Comparison

Turns out this is basic optimization.

By traveling 10 mph over the limit of 55, one would arrive 50 seconds quicker, risking a $113 fine and a 10–15 minute time delay if pulled over.

+ 5 mph at 30 mph has a greater effect on time than at 55 mph

It may seem counter-intuitive but the difference of increasing speed at lower rates has a greater effect on time than increasing speed at higher rates.

For instance, traveling without stopping for 5 miles at 30 mph equates to 10 minutes of travel time, whereas 5 miles at 35 mph equates to 8 minutes and 34 seconds.

This means you’ll arrive at your destination 1 minute and 26 seconds faster than if you were to drive 30mph. Compare this to increasing speed from 55 mph to 60 mph. We found this would only save 27 seconds.

Of course fines are adjusted accordingly and many states have higher penalties for speeding under 40mph.

So is it beneficial to speed? Crunch the numbers, consider the risks and you decide!

Next Lesson: Understanding Exponent Properties

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