More Math Problems Involving Distance, Rate And Time

Distance Rate Time Charts

Yesterday we discussed how to solve a certain type of rate problem very quickly simply by plugging numbers into a simple formula. In case you are interested you can find that post here: Xiggi’s formula.

In yesterday’s post we learned that Xiggi’s formula can be used to find an average rate when two individual rates for the same distance are known.

But what if we want to find the distance or time instead?

If the two distances are the same, then Xiggi’s formula can still be used but the solution will require a little more work. In this case however Xiggi’s formula is probably not the most efficient way to solve the problem.

Also, what if we want to solve a rate problem where the two (or more) distances are different?

Well in this case we certainly cannot use Xiggi’s formula. Another method will be needed.

In this blog post I would like to go over a method of solution that will work for any math problem involving distances, rates and times that show up on standardized tests such as the SAT, ACT and GRE. Today we will carefully go over how to set up a “distance = rate · time” chart.

Let’s start right away with an example.

Marco drove from home to work at an average speed of 50 miles per hour and returned home along the same route at an average speed of 46 miles per hour. If his total driving time for the trip was 4 hours, how many minutes did it take Marco to drive from work to home?

Let’s solve this problem by writing out a “distance = rate · time” chart.

d=rt chart2

Note that although we do not know either distance, we do know that they are the same, so we can call them both “d.” Also, since

distance = rate · time,

we have that

$\text{time} = \frac{\text{distance}}{\text{rate}}.$

We use this to get the first two entries in column three. The total time is given in the question. So we have

$\frac{d}{50} + \frac{d}{46}=4$

46d + 50d = 4 · 50 · 46

96d = 4 · 50 · 46

$d = \frac{4 \cdot 50 \cdot 46}{96}$

We want the time it takes Marco to drive from work to home, that is we want to compute d/46.

In hours, this is equal to

$\frac{d}{46} = \frac{4 \cdot 50}{96}$

To convert to minutes we multiply by 60.

$\frac{d}{46} = \frac{4 \cdot 50 \cdot 60}{96} = \bf{125} \text{ minutes}$

Let’s also try to solve this problem using Xiggi’s formula.

We have

$\text{Average speed} = \frac{2(50)(46)}{50+46} = \frac{575}{12}.$

$\text{So total round trip distance} = r\cdot t = \frac{575}{12} \cdot 4 = \frac{575}{3}.$

$\text{So distance from work to home} = \frac{575}{3} \div 2 = \frac{575}{6}.$

$\text{So time from work to home} = \frac{d}{r} = \frac{575}{6} \div 46 = \frac{25}{12}.$

Finally to convert from hours to minutes we multiply by 60 and get 125.

Notice that this particular problem was not so straightforward to solve using Xiggi’s formula. Compare this to the problem in yesterday’s blog post. Do you see the difference? In last week’s question we were being asked to find a rate, whereas in this question we were asked to find a time.

Toward the end of yesterday’s post you were asked to solve the following problem:

Jason ran a race of 1600 meters in two laps of equal distance. His average speeds for the first and second laps were 11 meters per second and 7 meters per second, respectively. What was his average speed for the entire race, in meters per second?