### What are distance word problems?

**Distance word problems** are a common type of algebra word problems. They involve a scenario in which you need lớn figure out how **fast**, how **far**, or how **long** one or more objects have sầu traveled. These are often called **train problems** because one of the most famous types of distance problems involves finding out when two trains heading toward each other cross paths.

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In this lesson, you'll learn how khổng lồ solve sầu train problems & a few other comtháng types of distance problems. But first, let's look at some basic principles that apply to lớn **any** distance problem.

There are three basic aspects to lớn movement & travel: **distance**, **rate**, & **time**. To understand the difference ahy vọng these, think about the last time you drove sầu somewhere.

This diagram is a start lớn understanding this problem, but we still have sầu to figure out what lớn bởi vì with the numbers for **distance**, **rate**, và **time**. To keep traông chồng of the information in the problem, we'll phối up a table. (This might seem excessive now, but it's a good habit for even simple problems & can make solving complicated problems much easier.) Here's what our table looks like:

d | 65 | 2.5 |

We can put this information inlớn our formula: **distance = rate ⋅ time**.

We can use the **distance = rate ⋅ time** formula to find the distance Lee traveled.

d = rt

The formula *d = rt* looks lượt thích this when we plug in the numbers from the problem. The unknown **distance** is represented with the variable *d*.

d = 65 ⋅ 2.5

To find *d*, all we have sầu khổng lồ vì is multiply 65 and 2.5. **65 ⋅ 2.5** equals 162.5.

d = 162.5

We have sầu an answer lớn our problem: *d* = 162.5. In other words, the distance Lee drove from his house to the zoo is 162.5 miles.

Be careful khổng lồ use the same **units of measurement** for rate and time. It's possible lớn multiply 65 miles per **hour** by 2.5 **hours** because they use the same unit: an **hour**. However, what if the time had been written in a different unit, lượt thích in **minutes**? In that case, you'd have to lớn convert the time into lớn hours so it would use the same unit as the rate.

In the problem we just solved we calculated for **distance**, but you can use the *d = rt* formula lớn solve sầu for **rate** & **time** too. For example, take a look at this problem:

After work, Janae walked in her neighborhood for a half hour. She walked a mile-and-a-half total. What was her average tốc độ in miles per hour?

We can picture Janae's walk as something lượt thích this:

And we can set up the information from the problem we know like this:

distanceratetime1.5 | r | 0.5 |

The table is repeating the facts we already know from the problem. Janae walked **one-and-a-half miles** or 1.5 miles in a half hour, or 0.5 hours.

As always, we start with our formula. Next, we'll fill in the formula with the information from our table.

d = rt

The rate is represented by *r* because we don't yet know how fast Janae was walking. Since we're solving for *r*, we'll have lớn get it alone on one side of the equation.

1.5 = r ⋅ 0.5

Our equation calls for *r* to lớn be **multiplied** by 0.5, so we can get *r* alone on one side of the equation by **dividing** both sides by 0.5: **1.5 / 0.5 =** 3.

3 = r

*r* = 3, so 3 is the answer khổng lồ our problem. Janae walked **3** miles per hour.

In the problems on this page, we solved for **distance** & **rate** of travel, but you can also use the travel equation to solve for **time**. You can even use it khổng lồ solve certain problems where you're trying to lớn figure out the distance, rate, or time of two or more moving objects. We'll look at problems lượt thích this on the next few pages.

### Two-part và round-trip problems

Do you know how lớn solve sầu this problem?

Bill took a trip lớn see a friend. His friover lives 225 miles away. He drove in town at an average of 30 mph, then he drove sầu on the interstate at an average of 70 mph. The trip took three-and-a-half hours total. How far did Bill drive sầu on the interstate?

This problem is a classic **two-part trip problem **because it's asking you to lớn find information about one part of a two-part trip. This problem might seem complicated, but don't be intimidated!

You can solve it using the same tools we used lớn solve sầu the simpler problems on the first page:

The**travel equation**

*d = rt*A

**table**khổng lồ keep trachồng of important information

Let's start with the **table**. Take another look at the problem. This time, the information relating khổng lồ **distance**, **rate**, and **time** has been underlined.

Bill took a trip khổng lồ see a friover. His friover lives 225 miles away. He drove sầu in town at an average of 30 mph, then he drove sầu on the interstate at an average of 70 mph. The trip took three-and-a-half hours total. How far did Bill drive on the interstate?

If you tried lớn fill in the table the way we did on the last page, you might have sầu noticed a problem: There's **too much** information. For instance, the problem contains **two** rates—30 **mph **and 70 **mph**. To include all of this information, let's create a table with an extra row. The top row of numbers and variables will be labeled **in town**, và the bottom row will be labeled **interstate**.

in town | 30 | ||

interstate | 70 |

We filled in the rates, but what about the **distance** & **time**? If you look baông xã at the problem, you'll see that these are the **total** figures, meaning they include both the time in town & on the interstate. So the** total distance ** is 225. This means this is true:

Interstate distance + in-town distance = Total distance

Together, the interstate distance & in-town distance are equal lớn the** total** distance. See?

In any case, we're trying lớn find out how far Bill drove sầu on the **interstate**, so let's represent this number with *d*. If the interstate distance is *d*, it means the in-town distance is a number that equals the total, 225, when **added** to lớn *d*. In other words, it's equal to lớn 225 - *d*.

We can fill in our chart like this:

distanceratetimein town | 225 - d | 30 | |

interstate | d | 70 |

We can use the same technique lớn fill in the **time** column. The total time is 3.5 **hours**. If we say the time on the interstate is *t*, then the remaining time in town is equal to 3.5 - *t*. We can fill in the rest of our chart.

in town | 225 - d | 30 | 3.5 - t |

interstate | d | 70 | t |

Now we can work on solving the problem. The main difference between the problems on the first page and this problem is that this problem involves **two **equations. Here's the one for **in-town travel**:

225 - d = 30 ⋅ (3.5 - t)

And here's the one for** interstate travel**:

d = 70t

If you tried khổng lồ solve either of these on its own, you might have sầu found it impossible: since each equation contains two unknown variables, they can't be solved on their own. Try for yourself. If you work either equation on its own, you won't be able to lớn find a numerical value for *d*. In order to find the value of *d*, we'll also have lớn know the value of *t*.

We can find the value of *t* in both problems by combining them. Let's take another look at our travel equation for interstate travel.

While we don't know the numerical value of *d*, this equation does tell us that *d* is equal lớn **70 t**.

d = 70t

Since **70 t** và

*are*

**d****equal**, we can replace

*with*

**d****70**. Substituting

*t***70**for

*t**in our equation for interstate travel won't help us find the value of*

**d***t*—all it tells us is that

**70**is equal to itself, which we already knew.

*t*70t = 70t

But what about our other equation, the one for in-town travel?

225 - d = 30 ⋅ (3.5 - t)

When we replace the *d* in that equation with **70 t**, the equation suddenly gets much easier to lớn solve sầu.

225 - 70t = 30 ⋅ (3.5 - t)

Our new equation might look more complicated, but it's actually something we can solve. This is because it only has one variable: *t*. Once we find *t*, we can use it to calculate the value of *d*—và find the answer lớn our problem.

To simplify this equation và find the value of *t*, we'll have to lớn get the *t* alone on one side of the equals sign. We'll also have sầu lớn **simplify** the right side as much as possible.

225 - 70t = 30 ⋅ (3.5 - t)

Let's start with the right side: **30** times **(3.5 - t)** is 105 - 30

*t*.

225 - 70t = 105 - 30t

Next, let's cancel out the **225** next to lớn **70 t**. To do this, we'll subtract

**225**from both sides. On the right side, it means subtracting 225 from 105.

**105 - 225**is -1trăng tròn.

- 70t = -1trăng tròn - 30t

Our next step is to** group **like terms—rethành viên, our eventual goal is khổng lồ have *t* on the **left** side of the equals sign & a number on the **right**. We'll cancel out the -30*t* on the right side by **adding** **30 t** to both sides. On the right side, we'll add it to lớn -70

*t*.

**-70**is -40

*t*+ 30*t**t*.

- 40t = -120

Finally, to lớn get *t* on its own, we'll divide each side by its coefficient: -40. **-120 / - 40** is 3.

t = 3

So *t* is equal to 3. In other words, the **time** Bill traveled on the interstate is equal to **3 hours**. Rethành viên, we're ultimately trying to lớn find the **distanc****e** Bill traveled on the interstate. Let's look at the **interstate** row of our chart again & see if we have enough information to lớn find out.

interstate | d | 70 | 3 |

It looks like we do. Now that we're only missing one variable, we should be able lớn find its value pretty quickly.

To find the distance, we'll use the travel formula **distance = rate ⋅ time**.

d = rt

We now know that Bill traveled on the interstate for 3 hours at 70 **mph**, so we can fill in this information.

d = 3 ⋅ 70

Finally, we finished simplifying the right side of the equation. **3 ⋅ 70** is 210.

d = 210

So * d* = 210. We have the answer khổng lồ our problem! The distance is 210. In other words, Bill drove sầu

**210 miles**on the interstate.

Solving a round-trip problem

It might have seemed lượt thích it took a long time lớn solve sầu the first problem. The more practice you get with these problems, the quicker they'll go. Let's try a similar problem. This one is called a **round-trip problem** because it describes a round trip—a trip that includes a return journey. Even though the trip described in this problem is slightly different from the one in our first problem, you should be able to lớn solve it the same way. Let's take a look:

Eva drove lớn work at an average tốc độ of 36 mph. On the way home page, she hit traffic and only drove sầu an average of 27 mph. Her total time in the car was 1 hour and 45 minutes, or 1.75 hours. How far does Eva live from work?

If you're having trouble understanding this problem, you might want lớn visualize Eva's commute lượt thích this:

As always, let's start by filling in a table with the important information. We'll make a row with information about her trip **khổng lồ work** and **from work**.

**1.75 - *** t* to describe the trip from work. (Rethành viên, the

**total**travel time is

**1.75 hours**, so the time

**to**work and

**from**work should equal 1.75.)

From our table, we can write two equations:

The trip**khổng lồ work**can be represented as

*d*= 36

*t*.The trip

**from work**can be represented as

*d*= 27 (1.75 -

*t*).

In both equations, *d* represents the total distance. From the diagram, you can see that these two equations are **equal** to each other—after all, Eva drives the **same distance lớn & from work**.

Just lượt thích with the last problem we solved, we can solve this one by **combining** the two equations.

We'll start with our equation for the trip **from work**.

d = 27 (1.75 - t)

Next, we'll substitute in the value of *d* from our **lớn work** equation, ** d = 36t**. Since the value of

*d*is 36

*t*, we can replace any occurrence of

*d*with 36

*t*.

36t = 27 (1.75 - t)

Now, let's simplify the right side. **27 ⋅(1.75 - t)** is 47.25.

36t = 47.25 - 27t

Next, we'll cancel out **-27 t** by

**adding**27

*t*to lớn both sides of the equation.

**36**is 63

*t*+ 27*t**t*.

63t = 47.25

Finally, we can get *t* on its own by dividing both sides by its coefficient: 63.** 47.25 / 63** is .75.

t = .75

*t* is equal to lớn .75. In other words, the **time** it took Eva to lớn drive lớn work is **.75 hours**. Now that we know the value of *t*, we'll be able to lớn can find the **distance** khổng lồ Eva's work.

If you guessed that we were going to use the **travel equation** again, you were right. We now know the value of two out of the three variables, which means we know enough lớn solve sầu our problem.

d = rt

First, let's fill in the values we know. We'll work with the numbers for the trip **to lớn work**. We already knew the **rate**: 36. And we just learned the **time**: .75.

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d = 36 ⋅ .75

Now all we have sầu khổng lồ vị is simplify the equation: **36 ⋅ .75** = 27.

d = 27

* d* is equal lớn 27. In other words, the

**distance**lớn Eva's work is

**27 miles**. Our problem is solved.

### Intersecting distance problems

An intersecting distance problem is one where two things are moving **toward** each other. Here's a typical problem:

Pawnee & Springfield are 420 miles apart. A train leaves Pawnee heading to lớn Springfield at the same time a train leaves Springfield heading to Pawnee. One train is moving at a speed of 45 mph, và the other is moving 60 mph. How long will they travel before they meet?

This problem is asking you to lớn 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 lớn 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** & 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 tốc độ of 45 mph, & 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 tốc độ of each train. We'll label them **fast train** & **slow train**. The fast train goes 60 **mph**. The slow train goes only 45 **mph**.

We can also put this information into lớn a table:

distanceratetimelớn work | d | 36 | t |

from work | d | 27 | 1.75 - t |

Notice that the **distance** is the same on both rows. Eva drove the same way and same distance lớn và from work. The **rate** is the speed she drove sầu during each part of the trip. For the time, we used *t* to lớn st& for the time it took lớn drive sầu **to** work, &

fast train | 60 | ||

slow train | 45 |

**total**distance. In order khổng lồ 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 4đôi mươi - *d*.

fast train | d | 60 | |

slow train | 4trăng tròn - 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*.

fast train | d | 60 | t |

slow train | 420 - d | 45 | t |

The table gives us **two** equations: *d* = 60*t* & 420 - *d* = 45*t*. Just lượt thích 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 khổng lồ 60*t*.

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.

4trăng tròn - d = 45t

Because we know that *d* is equal khổng lồ 60*t*, we can replace the *d* in this equation with **60 t**. Now we have sầu an equation we can solve.

4trăng tròn - 60t = 45t

To solve this equation, we'll need to lớn get *t* & its coefficients on one side of the equals sign và any other numbers on the other. We can start by canceling out the -60*t* on the left by **adding** **60 t** lớn both sides.

**45**is 105

*t*+ 60*t**t*.

4đôi mươi = 105t

Now we just need lớn get rid of the coefficient next to lớn *t*. We can vày this by dividing both sides by 105. **420 / 105** is 4.

4 = t

*t *= 4. In other words, the **time** it takes the trains khổng lồ meet is **4 hours**. Our problem is solved!

If you want khổng lồ 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 lớn the one we just solved. See if you can solve sầu it on your own. When you're finished, scroll down lớn see the answer và an explanation.

Jon và Dani live sầu 270 miles apart. One day, they decided lớn drive sầu toward each other và hang out wherever they met. Jon drove sầu an average of 65 mph, and Dani drove an average of 70 mph. How long did they drive sầu before they met up?

Problem 1 answerHere's practice problem 1:

Jon và Dani live sầu 270 miles apart. One day, they decided to lớn drive sầu toward each other & hang out wherever they met. Jon drove sầu an average of 65 mph, và Dani drove sầu 70 mph. How long did they drive sầu 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 lượt thích this:

distanceratetimeJon | d | 65 | t |

Dani | 270 - d | 70 | t |

Here's how we filled in the chart:

**Distance:**Together, Dani and Jon will have covered the

**total distance**between them by the time they meet up. That's 270. Jon's distance is represented by

*d*, so Dani's distance is 270 -

*d*.

**Rate:**The problem tells us Dani và Jon's speeds. Dani drives 65

**mph**, and Jon drives 70

**mph**.

**Time:**Because Jon & Dani drive the same amount of time before they meet up, both of their travel times are represented by

*t*.

Now we have sầu two equations. The equation for Jon's travel is *d* = 65*t*. The equation for Dani's travel is 270 - *d* = 70*t*. To solve sầu this problem, we'll need lớn **combine** them.

The equation for Jon tells us that *d* is equal to lớn 65*t*. This means we can combine the two equations by replacing the *d* in Dani's equation with 65*t*.

270 - 65t = 70t

Let's get *t* on one side of the equation và a number on the other. The first step khổng lồ doing this is to lớn get rid of -65*t* on the left side. We'll cancel it out by **adding** 65*t* khổng lồ both sides: **70 t + 65t** is 135

*t*.

270 = 135t

All that's left to do is lớn get rid of the 135 next to the *t*. We can vày this by dividing both sides by 135: **270 / 135** is 2.

2 = t

That's it. *t* is equal to lớn 2. We have sầu the answer to lớn our problem: Dani and Jon drove **2 hours** before they met up.

### Overtaking distance problems

The final type of distance problem we'll discuss in this lesson is a problem in which one moving object **overtakes**—or **passes**—another. Here's a typical overtaking problem:

The Hill family và the Platter family are going on a road trip. The Hills left 3 hours before the Platters, but the Platters drive an average of 15 mph faster. If it takes the Platter family 13 hours to lớn catch up with the Hill family, how fast are the Hills driving?

You can picture the moment the Platter family left for the road trip a little like this:

The problem tells us that the Platter family will catch up with the Hill family in 13 hours and asks us lớn use this information khổng lồ find the Hill family's **rate**. Like some of the other problems we've sầu solved in this lesson, it might not seem like we have sầu enough information khổng lồ solve sầu this problem—but we bởi vì. Let's start making our chart. The **distance** can be *d* for both the Hills & the Platters—when the Platters catch up with the Hills, both families will have sầu driven the exact same distance.

the Hills | d | ||

the Platters | d |

Filling in the **rate** and **time** will require a little more thought. We don't know the rate for either family—rethành viên, that's what we're trying khổng lồ find out. However, we vị know that the Platters drove sầu 15 mph **faster** than the Hills. This means if the Hill family's rate is *r*, the Platter family's rate would be *r* + 15.

the Hills | d | r | |

the Platters | d | r + 15 |

Now all that's left is the time. We know it took the Platters **13 hours** khổng lồ catch up with the Hills. However, remember that the Hills left **3 hours** earlier than the Platters—which means when the Platters caught up, they'd been driving **3 hours more** than the Platters. **13 + 3** is 16, so we know the Hills had been driving **16 hours** by the time the Platters caught up with them.

the Hills | d | r | 16 |

the Platters | d | r + 15 | 13 |

Our chart gives us two equations. The Hill family's trip can be described by *d* = *r *⋅ 16. The equation for the Platter family's trip is *d* = (*r* + 15) ⋅ 13. Just lượt thích with our other problems, we can **combine** these equations by replacing a variable in one of them.

The Hill family equation already has the value of *d* equal to *r* ⋅ 16. So we'll replace the *d* in the Platter equation with ** r ⋅ 16**. This way, it will be an equation we can solve.

r ⋅ 16 = (r + 15) ⋅ 13

First, let's simplify the right side: *r* ⋅ 16 is 16*r*.

16r = (r + 15) ⋅ 13

Next, we'll simplify the right side and multiply (*r* + 15) by 13.

16r = 13r + 195

We can get both *r *and their coefficients on the left side by **subtracting** 13*r* from 16*r* : **16 r** -

**13**is 3

*r**r*.

3r = 195

Now all that's left lớn vì is get rid of the 3 next lớn the *r*. To vì chưng this, we'll divide both sides by 3: **195 / 3** is 65.

r = 65

So there's our answer: *r* = 65. The Hill family drove an average of** 65 mph**.

You can solve sầu any overtaking problem the same way we solved this one. Just remember to pay special attention when you're setting up your chart. Just lượt thích the Hill family did in this problem, the person or vehicle who started moving **first** will always have a **greater** travel time.

Try solving this problem. It's similar to lớn the problem we just solved. When you're finished, scroll down to lớn see the answer & an explanation.

A train moving 60 mph leaves the station at noon. An hour later, a train moving 80 mph leaves heading the same direction on a parallel trachồng. What time does the second train catch up to the first?

Problem 2 answerHere's practice problem 2:

A train moving 60 mph leaves the station at noon. An hour later, a train moving 80 mph leaves heading the same direction on a parallel traông xã. What time does the second train catch up khổng lồ the first?

Answer: **4 p.m.**

To solve sầu this problem, start by making a chart. Here's how it should look:

distanceratetimefast train | d | 80 | t |

slow train | d | 60 | t + 1 |

Here's an explanation of the chart:

**Distance:**Both trains will have traveled the same distance by the time the fast train catches up with the slow one, so the distance for both is

*d*.

**Rate:**The problem tells us how fast each train was going. The fast train has a rate of 80

**mph**, and the slow train has a rate of 60

**mph**.

**Time:**We'll use

*t*to represent the fast train's travel time before it catches up. Because the slow train started an hour before the fast one, it will have been traveling one hour more by the time the fast train catches up. It's

*t*+ 1.

Now we have two equations. The equation for the fast train is *d* = 80*t*. The equation for the slow train is *d* = 60 (*t* + 1). To solve sầu this problem, we'll need lớn **combine** the equations.

The equation for the fast train says *d* is equal khổng lồ 80*t*. This means we can combine the two equations by replacing the *d* in the slow train's equation with 80*t*.

80t = 60 (t + 1)

First, let's simplify the right side of the equation: **60 ⋅ ( t + 1)** is 60

*t*+ 60.

80t = 60t + 60

To solve sầu the equation, we'll have to lớn get *t* on one side of the equals sign and a number on the other. We can get rid of 60*t* on the right side by **subtracting** 60*t* from both sides: 80*t* - 60*t* is 20*t*.

20t = 60

Finally, we can get rid of the 20 next lớn *t* by dividing both sides by đôi mươi. **60 **divided by** 20** is 3.

t = 3

So *t* is equal khổng lồ 3. The fast train traveled for **3 hours**. However, it's not the answer to lớn our problem. Let's look at the original problem again. Pay attention to the last sentence, which is the question we're trying to lớn answer.

A train moving 60 mph leaves the station at noon. An hour later, a train moving 80 mph leaves heading the same direction on a parallel traông chồng. What time does the second train catch up to lớn the first?

Our problem doesn't ask how **long** either of the trains traveled. It asks **what time** the second train catches up with the first.

The problem tells us that the slow train left at noon và the fast one left an hour later. This means the fast train left at **1 p.m**. From our equations, we know the fast train traveled 3 **hours**. **1 + 3** is 4, so the fast train caught up with the slow one at **4 p.m**. The answer to lớn the problem is 4 p.m.