Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
Solve for Distance
How fast can you run? How far does your car go on a tank of gas? If you keep going on your bike at a constant speed, how long will it take to get to the next town?
These are the problems that vex us. Well, OK, these are problems that vex me, particularly on long drives in big, open states like Wyoming, where I really want to know how quickly I can get from point A to point B.
Fortunately, there are formulas we can use to solve problems like these. What's better is that they're simple enough that you might even be able to do them in your head while speeding along the interstate and eating french fries, though hopefully not also using your phone. That's totally dangerous.
Let's start with the distance formula. The standard distance formula, which is also known as the uniform rate formula, is d = rt. D stands for distance, r for rate and t for time.
How can we use this? Well, what if we know the speed of something and how long it's moving, but not how far it goes? Here's a sample problem:
Sarah runs at an average rate of 7.5 miles per hour for 2 hours. How far does she go?
Let's start by identifying what we know from our formula. Our r, or rate, is 7.5 miles per hour. Our t, or time, is 2 hours. So d = 7.5 * 2. Note that we're multiplying miles per hour times hours. If we were to write this properly, it would be 7.5miles/1 hour * 2 hours. The hours cancel out, leaving us with just miles.
Since we're trying to find distance, it makes sense that our answer will be in miles. But always be sure that you're using the same language. If you're given time in minutes, make sure your rate is in minutes. If not, you'll need to convert one or the other. If you're given miles per hour and the question asks about kilometers or parsecs or something like that, well, then your conversion might get a little messy.
But here we just want to know how many miles Sarah goes. Again, we have d = 7.5 * 2. That means d = 15. So Sarah ran 15 miles. That's awesome!
Solve for Rate
The distance formula is nicely versatile. It's not all about how far Sarah can run, though that's very impressive. We can rewrite the distance formula if, instead of distance, we're solving for something else, like the rate. Here's a rate problem:
A train travels 438 miles in 6 hours. What is its average speed?
Let's look at our distance formula: d = rt. Here we know our d is 438 and our t is 6. We want to know r. So we can rewrite the formula as r = d/t. All we did is divide by t to move the t over.
So r = 438/6. And that's 73. So, this train traveled at an average speed of 73 mph. That's a pretty fast train.
Solve for Time
That's solving for r; what about solving for t? That brings us to distance formula iteration #3: t = d/r. Here's a problem where this is useful:
Mike walks at a rate of 3 miles per hour. If he needs to walk 26.2 miles, how long will his trip last?
So Mike's walking a marathon. Why? I don't know. That's between Mike and his soon-to-be-tired feet. But we can figure this out. If t = d/r, then we have t = 26.2/3. That's 8.73. .73 is about 44 minutes. So, Mike will be walking for 8 hours and 44 minutes. That's a long walk.
Distance Plus Distance
With this next one, we're going to try something different: distance plus distance. Here's the problem:
Sam drives 85 mph for 3 hours. He gets pulled over for speeding, so he drives 55 mph for the next 2 hours. What's the total distance Sam traveled?
Okay; to find the distance plus distance, we need to tackle this problem in two parts. In the first part of his trip, Sam travels at a rate of 85 mph for a time of 3 hours. With d = rt, we have d = 85 * 3. That's 255 miles.
In the second part, Sam travels at a rate of 55 mph for 2 hours. That's d = 55 * 2. So he went 110 miles in the second part. Now we add the two parts, 255 + 110, to get 365. So, Sam went a total of 365 miles.
Distance and Fuel
There's one other type of distance problem to talk about here: distance and fuel.
Mike is embarking on a road trip. He starts in Maine, which they say is the way life should be. That's true in July, but Mike is leaving in February. He wants to go to southern California, where it's always summer. Here's a word problem based on the trip:
A car uses 4 tanks of fuel to travel 1300 miles. At this rate, how many tanks of fuel would be required for a trip of 3000 miles?
So 4 tanks of fuel to go 1300 miles. That only gets Mike to Iowa. What'll it take to get to SoCal? We don't need a formula here. Let's just set up two ratios. First, 4 tanks goes 1300 miles, so we have 4 tanks/1300 miles. Next, there's x number of tanks to go 3000 miles, so x tanks/3000 miles.
Let's set these equal to each other and cross multiply. That's 1300x = 3000*4. 3000*4 is 12,000. Divide that by 1300, and we get 9.23. So, Mike will need about 9 and a quarter tanks of gas to escape the wintry wonderland of Maine for the palm trees of California.
To summarize, we looked at the distance formula, which is also known as the uniform rate formula. The traditional version of this formula is d = rt, where d is distance, r is rate and t is time.
This formula can be rewritten as r = d/t to solve for the rate or t = d/r to solve for the time.
We also looked at the relationship between distance and fuel. To solve problems that compare the distance traveled to the amount of fuel used, we set up an equation involving ratios, then cross multiplied to solve.
By the end of this lesson you should be able to:
- Recall the distance formula
- Calculate for distance, time, rate, or fuel usage in travel math problems
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