In algebra distance word problems are a struggle for some. Use this free lesson to help you learn how to solve distance word problems.
An intersecting distance problem is one where two things are moving toward each other. Here's a typical problem:
Pawnee and Springfield are 420 miles apart. A train leaves Pawnee heading to Springfield at the same time a train leaves Springfield heading to Pawnee. One train is moving at a speed of 45 mph, and the other is moving 60 mph. How long will they travel before they meet?
This problem is asking you to calculate how long it will take these two trains moving toward each other to cross paths. This might seem confusing at first. Even though it's a real-world situation, it can be difficult to imagine distance and motion abstractly. This diagram might help you get a sense of what this situation looks like:
If you're still confused, don't worry! You can solve this problem the same way you solved the two-part problems on the last page. You'll just need a chart and the travel formula.
Pawnee and Springfield are 420 miles apart. A train leaves Pawnee heading toward Springfield at the same time a train leaves Springfield heading toward Pawnee. One train is moving at a speed of 45 mph, and the other is moving 60 mph. How long will they travel before they meet?
Let's start by filling in our chart. Here's the problem again, this time with the important information underlined. We can start by filling in the most obvious information: rate. The problem gives us the speed of each train. We'll label them fast train and slow train. The fast train goes 60 mph. The slow train goes only 45 mph.
We can also put this information into a table:
We don't know the distance each train travels to meet the other yet—we just know the total distance. In order to meet, the trains will cover a combined distance equal to the total distance. As you can see in this diagram, this is true no matter how far each train travels.
This means that—just like last time—we'll represent the distance of one with d and the distance of the other with the total minus d. So the distance for the fast train will be d, and the distance for the slow train will be 420 - d.
|slow train||420 - d||45|
Because we're looking for the time both trains travel before they meet, the time will be the same for both trains. We can represent it with t.
|slow train||420 - d||45||t|
The table gives us two equations: d = 60t and 420 - d = 45t. Just like we did with the two-part problems, we can combine these two equations.
The equation for the fast train isn't solvable on its own, but it does tell us that d is equal to 60t.
d = 60t
The other equation, which describes the slow train, can't be solved alone either. However, we can replace the d with its value from the first equation.
420 - d = 45t
Because we know that d is equal to 60t, we can replace the d in this equation with 60t. Now we have an equation we can solve.
420 - 60t = 45t
To solve this equation, we'll need to get t and its coefficients on one side of the equals sign and any other numbers on the other. We can start by canceling out the -60t on the left by adding 60t to both sides. 45t + 60t is 105t.
420 = 105t
Now we just need to get rid of the coefficient next to t. We can do this by dividing both sides by 105. 420 / 105 is 4.
4 = t
t = 4. In other words, the time it takes the trains to meet is 4 hours. Our problem is solved!
If you want to be sure of your answer, you can check it by using the distance equation with t equal to 4. For our fast train, the equation would be d = 60 ⋅ 4. 60 ⋅ 4 is 240, so the distance our fast train traveled would be 240 miles. For our slow train, the equation would be d = 45 ⋅ 4. 45 ⋅ 4 is 180, so the distance traveled by the slow train is 180 miles.
Remember how we said the distance the slow train and fast train travel should equal the total distance? 240 miles + 180 miles equals 420 miles, which is the total distance from our problem. Our answer is correct.
Here's another intersecting distance problem. It's similar to the one we just solved. See if you can solve it on your own. When you're finished, scroll down to see the answer and an explanation.
Jon and Dani live 270 miles apart. One day, they decided to drive toward each other and hang out wherever they met. Jon drove an average of 65 mph, and Dani drove an average of 70 mph. How long did they drive before they met up?
Here's practice problem 1:
Jon and Dani live 270 miles apart. One day, they decided to drive toward each other and hang out wherever they met. Jon drove an average of 65 mph, and Dani drove 70 mph. How long did they drive before they met up?
Answer: 2 hours.
Let's solve this problem like we solved the others. First, try making the chart. It should look like this:
|Dani||270 - d||70||t|
Here's how we filled in the chart:
Now we have two equations. The equation for Jon's travel is d = 65t. The equation for Dani's travel is 270 - d = 70t. To solve this problem, we'll need to combine them.
The equation for Jon tells us that d is equal to 65t. This means we can combine the two equations by replacing the d in Dani's equation with 65t.
270 - 65t = 70t
Let's get t on one side of the equation and a number on the other. The first step to doing this is to get rid of -65t on the left side. We'll cancel it out by adding 65t to both sides: 70t + 65t is 135t.
270 = 135t
All that's left to do is to get rid of the 135 next to the t. We can do this by dividing both sides by 135: 270 / 135 is 2.
2 = t
That's it. t is equal to 2. We have the answer to our problem: Dani and Jon drove 2 hours before they met up.