## Part 1: Introduction to Speed, Distance, and Time

The concepts of speed, distance, and time are crucial to solving many problems in the GRE. Understanding the relationships between these three variables will help you tackle a wide range of word problems.

### 1. Definitions:

• Speed: Speed is a measure of how fast an object is moving. It is defined as the distance traveled per unit of time. The most commonly used units for speed are meters per second (m/s) and kilometers per hour (km/h).
• Distance: Distance is a measure of how far an object has traveled. It is usually measured in meters (m) or kilometers (km).
• Time: Time is the duration of an event or the interval between two events. It is typically measured in seconds (s), minutes (min), or hours (h).

### 2. Relationship between Speed, Distance, and Time:

The relationship between speed, distance, and time can be represented by the following formula:

Speed = Distance / Time

Alternatively, you can rearrange the formula to find the distance or time:

Distance = Speed × Time Time = Distance / Speed

### 3. Conversion between units:

Sometimes, you may need to convert between different units of speed, distance, or time. The most common conversions are:

• 1 km = 1000 m
• 1 hour = 60 minutes
• 1 minute = 60 seconds

To convert from kilometers per hour (km/h) to meters per second (m/s):

Speed (m/s) = Speed (km/h) × (1000 m/km) / (3600 s/h)

### 4. Tips for solving Speed, Distance, and Time word problems:

• Read the problem carefully and identify the given information.
• Determine the relationship between the given information and the unknown variable(s).
• Use the appropriate formula to solve for the unknown variable(s).
• Convert units if necessary.
• Check your answer to ensure it makes sense in the context of the problem.

In the following parts, we will discuss various types of word problems involving speed, distance, and time, along with strategies to solve them effectively.

## Part 2: Types of Speed, Distance, and Time Word Problems

There are several types of word problems involving speed, distance, and time that you may encounter on the GRE. In this Part, we will discuss different types of word problems and provide examples for each.

### 1. Constant Speed Problems:

In constant speed problems, the speed of an object remains the same throughout the entire journey. You can use the basic formula: Speed = Distance / Time to solve these problems.

Example: A car travels at a constant speed of 60 km/h for 2 hours. How far does the car travel?

Solution: Using the formula, Distance = Speed × Time, we can find the distance traveled by the car: Distance = 60 km/h × 2 h = 120 km

### 2. Average Speed Problems:

In average speed problems, the object travels at different speeds during its journey. The average speed is the total distance traveled divided by the total time taken.

Example: A cyclist travels 30 km at a speed of 15 km/h and then travels another 30 km at a speed of 10 km/h. What is the cyclist’s average speed for the entire journey?

Solution: First, find the time taken for each Part of the journey: Time1 = 30 km / 15 km/h = 2 h Time2 = 30 km / 10 km/h = 3 h

Next, find the total distance and total time: Total Distance = 30 km + 30 km = 60 km Total Time = 2 h + 3 h = 5 h

Finally, find the average speed. Average Speed = Total Distance / Total Time = 60 km / 5 h = 12 km/h

### 3. Relative Speed Problems:

In relative speed problems, two objects are moving relative to each other. The relative speed is the difference between their individual speeds if they are moving in opposite directions or the sum of their individual speeds if they are moving in the same direction.

Example: Two trains are moving towards each other on parallel tracks. Train A is traveling at a speed of 60 km/h, and Train B is traveling at a speed of 80 km/h. What is their relative speed?

Solution: Since the trains are moving towards each other, their relative speed is the sum of their individual speeds: Relative Speed = 60 km/h + 80 km/h = 140 km/h

In the next Part, we will discuss more advanced types of word problems involving speed, distance, and time and provide strategies to solve them effectively.

## Part 3: Advanced Speed, Distance, and Time Word Problems

In this Part, we will discuss more advanced types of word problems involving speed, distance, and time, along with strategies to solve them effectively.

### 1. Problems involving time intervals:

In these problems, you need to find the time taken to complete a task or the time at which a certain event occurs.

Example: A car travels at a constant speed of 80 km/h for the first 3 hours of its journey, then increases its speed to 100 km/h for the remaining Part of the journey. If the car travels a total distance of 600 km, how long does the entire journey take?

Solution: First, find the distance traveled at 80 km/h: Distance1 = 80 km/h × 3 h = 240 km

Next, find the remaining distance to be traveled at 100 km/h: Distance2 = Total Distance – Distance1 = 600 km – 240 km = 360 km

Now, find the time taken to travel the remaining distance at 100 km/h: Time2 = Distance2 / 100 km/h = 360 km / 100 km/h = 3.6 h

Finally, find the total time taken for the journey: Total Time = Time1 + Time2 = 3 h + 3.6 h = 6.6 h

### 2. Problems involving multiple objects:

In these problems, two or more objects are traveling at different speeds, and you need to find their meeting point, the time taken to meet, or the distance between them.

Example: Two cars, A and B, start traveling towards each other from two points, 300 km apart, at the same time. Car A travels at a constant speed of 60 km/h, while Car B travels at a constant speed of 90 km/h. How long does it take for the cars to meet?

Solution: Since the cars are traveling towards each other, their relative speed is the sum of their individual speeds: Relative Speed = 60 km/h + 90 km/h = 150 km/h

Next, find the time taken for the cars to meet: Time = Distance / Relative Speed = 300 km / 150 km/h = 2 h

### 2. Problems involving changing speed or acceleration:

In these problems, the speed of an object changes during its journey, and you need to find the distance traveled, time taken, or final speed.

Example: A car accelerates from rest at a constant rate of 4 m/s² for 5 seconds. What is the final speed of the car?

Solution: Use the formula for acceleration: Final Speed = Initial Speed + (Acceleration × Time). Since the car starts from rest, the initial speed is 0: Final Speed = 0 + (4 m/s² × 5 s) = 20 m/s

In the final Part, we will discuss strategies for solving more complex speed, distance, and time word problems and provide additional examples to practice.

## Part 4: Strategies for Complex Speed, Distance, and Time Word Problems and Additional Examples

In this final Part, we will discuss strategies for solving more complex speed, distance, and time word problems, along with additional examples to practice.

### Strategies:

#### 1. Break down complex problems into smaller parts:

If a problem seems too complicated, try breaking it down into smaller, more manageable parts. Solve each part separately and then combine the results to find the final solution.

#### 2. Use diagrams or sketches:

Drawing a diagram or sketch can help you visualize the problem and better understand the relationships between the given information.

#### 3. Please keep track of units:

Make sure to keep track of units throughout the problem and convert them as needed. Always double-check that your final answer has the correct units.

4. Double-check your calculations: Make sure to double-check your calculations and ensure that your answer makes sense in the context of the problem.

Example 1: A cyclist travels a distance of 80 km in 4 hours, maintaining a constant speed. The cyclist then takes a 1-hour break before cycling another 40 km in 2 hours. What is the cyclist’s average speed for the entire journey, including the break?

Solution: First, find the total distance and total time taken, including the break: Total Distance = 80 km + 40 km = 120 km Total Time = 4 h + 1 h (break) + 2 h = 7 h

Next, find the average speed for the entire journey: Average Speed = Total Distance / Total Time = 120 km / 7 h ≈ 17.14 km/h

Example 2: Two trains start traveling towards each other at the same time from two cities 500 km apart. Train A travels at a constant speed of 60 km/h, and Train B travels at a constant speed of 100 km/h. How far has Train A traveled when the two trains meet?

Solution: First, find their relative speed: Relative Speed = 60 km/h + 100 km/h = 160 km/h

Next, find the time it takes for the two trains to meet: Time = Distance / Relative Speed = 500 km / 160 km/h = 25/8 h

Finally, find the distance traveled by Train A when the two trains meet: Distance A = Speed A × Time = 60 km/h × (25/8 h) = 25 × 15 km = 375 km

By mastering the strategies and practicing different types of word problems involving speed, distance, and time, you can improve your problem-solving skills and increase your chances of success on the GRE.

## Practice questions with an answer key.

###### Difficulty level: Hard

Question 1:

A car travels at a constant speed of 60 km/h for the first 150 km of its journey. It then stops for 30 minutes to refuel and continues the journey at a constant speed of 90 km/h for the remaining 270 km. What is the car’s average speed for the entire journey, including the refueling time? (Round to the nearest integer)

A) 64 km/h B) 68 km/h C) 70 km/h D) 75 km/h E) 78 km/h

To find the car’s average speed for the entire journey, we need to consider the total distance traveled and the total time taken.
For the first part of the journey, the car travels at a constant speed of 60 km/h for 150 km. The time taken for this part can be calculated using the formula: time = distance / speed. So, the time taken for the first part is 150 km / 60 km/h = 2.5 hours.
After refueling, the car continues the journey at a constant speed of 90 km/h for 270 km. The time taken for this part is 270 km / 90 km/h = 3 hours.
Since the car stops for 30 minutes to refuel, we need to add this time to the total time taken. 30 minutes is equal to 0.5 hours.
The total time taken for the journey is 2.5 hours + 3 hours + 0.5 hours = 6 hours.
To find the average speed, we divide the total distance traveled by the total time taken: average speed = total distance / total time. The total distance is 150 km + 270 km = 420 km.
Average speed = 420 km / 6 hours ≈ 70 km/h.
Rounding to the nearest integer, the car’s average speed for the entire journey, including the refueling time, is 70 km/h.

Question 2:

Two runners, A and B, start at the same point on a circular track with a circumference of 600 meters. Runner A runs at a constant speed of 8 m/s, while Runner B runs at a constant speed of 10 m/s. How many seconds will it take for Runner B to catch up with Runner A for the first time?

A) 60 s B) 75 s C) 120 s D) 150 s E) 300 s

To find the time it takes for Runner B to catch up with Runner A, we need to determine the time it takes for Runner B to cover the distance between them.

Let’s assume that Runner B catches up with Runner A after t seconds. During this time, Runner A would have covered a distance equal to its speed multiplied by the time: distance_A = speed_A * t = 8t meters.

The distance around the circular track is 600 meters, so when Runner B catches up with Runner A, they will have completed a certain number of laps. Since the track’s circumference is 600 meters, each lap corresponds to that distance.

The difference in the number of laps completed by the two runners when they meet is the number of laps Runner B has completed minus the number of laps Runner A has completed. This can be calculated as (distance_B – distance_A) / circumference, where distance_B is the distance covered by Runner B.

The distance covered by Runner B is given by speed_B * t = 10t meters.

The difference in the number of laps is then ((10t – 8t) / 600) = (2t / 600) = t / 300.

Since the difference in the number of laps must be a whole number (B is catching up with A for the first time), t must be a multiple of 300.

The smallest positive multiple of 300 is 300 itself. Therefore, t = 300 seconds.

So, it will take 300 seconds for Runner B to catch up with Runner A for the first time.

Question 3:

A cyclist travels uphill for 40 km at a constant speed of 12 km/h and then downhill for another 40 km at a constant speed of 20 km/h. What is the cyclist’s average speed for the entire 80 km journey?

A) 14 km/h B) 15 km/h C) 15.2 km/h D) 15.5 km/h E) 16 km/h

To find the average speed for the entire journey, we need to consider the total distance traveled and the total time taken.
The cyclist travels uphill for 40 km at a speed of 12 km/h. The time taken for this part can be calculated using the formula: time = distance / speed. So, the uphill part takes 40 km / 12 km/h = 3.33 hours.
The cyclist then travels downhill for another 40 km at a speed of 20 km/h. The time taken for this part is 40 km / 20 km/h = 2 hours.
The total distance traveled is 40 km + 40 km = 80 km.
The total time taken for the journey is 3.33 hours + 2 hours = 5.33 hours.
To find the average speed, we divide the total distance traveled by the total time taken: average speed = total distance / total time. The average speed is 80 km / 5.33 hours ≈ 15.02 km/h.
Rounding to the nearest tenth, the cyclist’s average speed for the entire 80 km journey is approximately 15.0 km/h.

Question 4:

Two trains start traveling towards each other from two cities 720 km apart. Train A leaves City A at a constant speed of 80 km/h, while Train B leaves City B 1 hour later at a constant speed of 120 km/h. How many kilometers from City A will the two trains meet?

A) 288 km B) 360 km C) 400 km D) 420 km E) 480 km

To determine the distance from City A, where the two trains will meet, we need to consider the relative speeds of the trains.
Train A is traveling at a speed of 80 km/h, while Train B is traveling at a speed of 120 km/h. The relative speed of the two trains is the sum of their individual speeds: 80 km/h + 120 km/h = 200 km/h.
Since Train B departs City B 1 hour after Train A, it will have traveled for 1 hour less when the two trains meet. Therefore, we need to find the time it takes for Train A and Train B to cover a combined distance of 720 km.
Using the formula: time = distance / speed, the time taken for the trains to meet is 720 km / 200 km/h = 3.6 hours.
Since Train A has been traveling for 3.6 hours at a constant speed of 80 km/h, the distance it has covered is 3.6 hours * 80 km/h = 288 km.
Therefore, the two trains will meet 288 km from City A.

Question 5:

A boat travels downstream on a river at a constant speed of 15 km/h relative to the water. The river has a current speed of 5 km/h. If the boat takes 3 hours to travel downstream, how long will it take the boat to travel the same distance upstream against the current?

A) 2.25 hours B) 3 hours C) 4.5 hours D) 6 hours E) 9 hours

To determine the time it takes for the boat to travel the same distance upstream against the current, we need to consider the relative velocities of the boat and the river.
When the boat is traveling downstream, its effective speed is the sum of its speed relative to the water and the speed of the current. So, the boat’s effective speed downstream is 15 km/h + 5 km/h = 20 km/h.
Given that the boat takes 3 hours to travel downstream, we can calculate the distance traveled using the formula: distance = speed × time. Therefore, the distance traveled downstream is 20 km/h × 3 hours = 60 km.
To find the time it takes for the boat to travel the same distance upstream against the current, we need to subtract the speed of the current from the boat’s speed relative to the water. So, the boat’s effective speed upstream is 15 km/h – 5 km/h = 10 km/h.
Using the formula: time = distance / speed, the time taken for the boat to travel upstream is 60 km / 10 km/h = 6 hours.
Therefore, the boat will take 6 hours to travel the same distance upstream against the current.
The correct answer is D) 6 hours.

###### Difficulty level: Challenging

Question 1:

A car travels due east for 3 hours at a constant speed of 80 km/h. It then turns north and travels at a constant speed of 100 km/h for another 2 hours. What is the car’s average speed for the entire journey?

A) 84 km/h B) 88 km/h C) 90 km/h D) 92 km/h E) 96 km/h

To find the average speed for the entire journey, we need to consider the total distance traveled and the total time taken.
The car travels due east for 3 hours at a constant speed of 80 km/h. The distance traveled in this part is speed multiplied by time: distance = speed × time = 80 km/h × 3 hours = 240 km.
Afterward, the car turns north and travels for 2 hours at a constant speed of 100 km/h. The distance traveled in this part is speed multiplied by time: distance = speed × time = 100 km/h × 2 hours = 200 km.
The total distance traveled is 240 km + 200 km = 440 km.
The total time taken for the journey is 3 hours + 2 hours = 5 hours.
To find the average speed, we divide the total distance traveled by the total time taken: average speed = total distance / total time = 440 km / 5 hours = 88 km/h.
Therefore, the car’s average speed for the entire journey is 88 km/h.
The correct answer is B) 88 km/h.

Question 2:

A train travels at a constant speed of 60 km/h for the first 2 hours of its journey. The train then accelerates at a constant rate of 2 km/h² for the next hour. What is the train’s average speed for the entire 3-hour journey?

A) 60 km/h B) 63 km/h C) 66 km/h D) 69 km/h E) 72 km/h

The cyclist travels due west at a speed of 20 km/h20km/h for 4 hours, so the distance covered in the westward direction is given by:

Distancewest=speed×time=20 km/h×4 hours=80 kmDistancewest​=speed×time=20km/h×4hours=80km

Afterward, the cyclist turns due south and travels at a speed of 30 km/h30km/h for 2 hours, covering a distance of:

Distancesouth=speed×time=30 km/h×2 hours=60 kmDistancesouth​=speed×time=30km/h×2hours=60km

To return directly to point A, the cyclist needs to travel a distance equal to the total distance covered in the westward and southward directions. Therefore, the cyclist needs to travel a distance of:

Distancetotal=Distancewest+Distancesouth=80 km+60 km=140 kmDistancetotal​=Distancewest​+Distancesouth​=80km+60km=140km

To calculate the time it will take for the cyclist to cover this distance at a speed of 40 km/h40km/h, we can use the formula:

Time=Distancetotalspeed=140 km40 km/h=3.5 hoursTime=speedDistancetotal​​=40km/h140km​=3.5hours

Therefore, it will take the cyclist 3.5 hours to travel back to point A at a constant speed of 40 km/h40km/h.

Please note that the given options in the previous response are incorrect, and the correct time is 3.5 hours, which corresponds to the closest option available.

Question 3:

A cyclist starts at point A and travels due west at a constant speed of 20 km/h for 4 hours. He then turns due south and travels at a constant speed of 30 km/h for another 2 hours. If the cyclist wants to return directly to point A, how long will it take him to travel back, assuming he maintains a constant speed of 40 km/h?

A) 3.6 hours B) 4 hours C) 4.2 hours D) 4.8 hours E) 5 hours

Question 4:

Two planes leave an airport at the same time, with one flying due north at a constant speed of 400 km/h and the other flying due east at a constant speed of 500 km/h. How far apart are the two planes after 3 hours of flight?

A) 600 km B) 900 km C) 1200 km D) 1500 km E) 1800 km

Question 5:

A boat travels 30 km upstream on a river with a current of 6 km/h in 3 hours. The boat then turns around and travels downstream for another 30 km. What is the boat’s speed in still water?

A) 12 km/h B) 14 km/h C) 16 km/h D) 18 km/h E) 20 km/h

###### Difficulty level: hard

Question 1:

Quantity A: The time it takes a car to travel 200 km at a constant speed of 80 km/h.

Quantity B: The time it takes a car to travel 300 km at a constant speed of 120 km/h

A) Quantity A is greater

B) Quantity B is greater

C) The two quantities are equal

D) The relationship cannot be determined from the information given

Answer: C) The two quantities are equal

Question 2:

A cyclist covers a distance of 240 km in two parts. In the first Part, he cycles at a speed of 30 km/h, and in the second Part, he cycles at a speed of 40 km/h.

Quantity A: The time taken for the first Part of the journey

Quantity B: The time taken for the second Part of the journey

A) Quantity A is greater

B) Quantity B is greater

C) The two quantities are equal

D) The relationship cannot be determined from the information given

Answer: D) The relationship cannot be determined from the information given

Question 3:

Two buses travel the same distance from City A to City B. Bus X travels at a constant speed of 60 km/h and takes 1 hour longer to complete the journey than Bus Y, which travels at a constant speed of 80 km/h. Quantity A: The distance between City A and City B

Quantity B: 480 km

A) Quantity A is greater

B) Quantity B is greater

C) The two quantities are equal

D) The relationship cannot be determined from the information given

Answer: C) The two quantities are equal

Question 4:

Two runners, A and B, run around a 400-meter circular track. Runner A completes the track in 80 seconds, while Runner B completes it in 100 seconds.

Quantity A: The distance covered by Runner A in the time it takes Runner B to complete 5 laps.

Quantity B: The distance covered by Runner B in the time it takes Runner A to complete 6 laps

A) Quantity A is greater

B) Quantity B is greater

C) The two quantities are equal

D) The relationship cannot be determined from the information given

Answer: A) Quantity A is greater

###### Difficulty level: challenging

Question 1:

A car travels from Town A to Town B at an average speed of 60 km/h and then returns to Town A at an average speed of 40 km/h. The round trip takes 5 hours.

Quantity A: The distance between Town A and Town B

Quantity B: 120 km

A) Quantity A is greater

B) Quantity B is greater

C) The two quantities are equal

D) The relationship cannot be determined from the information given

Answer: A) Quantity A is greater

Question 2:

A train travels from City A to City B at a constant speed of 80 km/h and returns to City A at a constant speed of 120 km/h. The entire round trip takes 6 hours.

Quantity A: The time it takes to travel from City A to City B

Quantity B: The time it takes to travel from City B to City A

A) Quantity A is greater

B) Quantity B is greater

C) The two quantities are equal

D) The relationship cannot be determined from the information given

Answer: A) Quantity A is greater

Question 3:

Two planes leave an airport at the same time. Plane A flies due north at a constant speed of 500 km/h, while Plane B flies due east at a constant speed of 400 km/h.

Quantity A: The distance between the two planes after 3 hours of flight Quantity B: 1,500 km

A) Quantity A is greater

B) Quantity B is greater

C) The two quantities are equal

D) The relationship cannot be determined from the information given

Answer: A) Quantity A is greater

Question 4:

A runner completes a 5,000-meter race in 20 minutes. The first half of the race is uphill, and the runner’s average speed for this portion is 80% of his average speed for the downhill portion. Quantity A: The time it takes the runner to complete the uphill portion of the race Quantity B: The time it takes the runner to complete the downhill portion of the race

A) Quantity A is greater

B) Quantity B is greater

C) The two quantities are equal

D) The relationship cannot be determined from the information given

Answer: A) Quantity A is greater

### Question types: Multiple Select (three or more answer choices; one or more correct answers)

###### Difficulty level: hard

Question 1:

A car travels a total distance of 360 km in 4 hours. The car travels at a constant speed of 60 km/h for the first Part of the journey and at a constant speed of 120 km/h for the second Part of the journey. Which of the following statements is true? Select all that apply.

A) The first Part of the journey takes 2 hours.

B) The first Part of the journey covers 180 km.

C) The second Part of the journey takes 2 hours.

D) The second Part of the journey covers 240 km.

Answer: B) 180 km, D) 240 km

Question 2:

A cyclist travels a distance of 60 km, half of which is uphill at an average speed of 20 km/h. The cyclist’s average speed for the downhill portion is 50% faster than the uphill speed. Which of the following statements is true? Select all that apply.

A) The cyclist takes 1.5 hours to travel uphill.

B) The cyclist takes 1 hour to travel downhill.

C) The total time for the journey is 3 hours.

D) The average speed for the entire journey is 30 km/h.

Answer: A) 1.5 hours uphill, B) 1 hour downhill

Question 3: Two buses leave a bus station at the same time. Bus A travels due north at a constant speed of 40 km/h, while Bus B travels due east at a constant speed of 60 km/h. Which of the following statements is true after 3 hours? Select all that apply.

A) The two buses are 120 km apart.

B) The two buses are 180 km apart.

C) Bus A has traveled a distance of 120 km.

D) Bus B has traveled a distance of 180 km.

Answer: B) 180 km apart, C) Bus A traveled 120 km, D) Bus B traveled 180 km

Question 4:

A boat travels upstream on a river at a constant speed of 15 km/h relative to the water. The river has a current of 5 km/h. The boat then turns around and travels downstream at the same constant speed relative to the water. Which of the following statements is true? Select all that apply.

A) The boat’s speed upstream relative to the ground is 10 km/h.

B) The boat’s speed downstream relative to the ground is 20 km/h.

C) The boat takes twice as long to travel a given distance upstream as it does downstream.

D) The boat takes half as long to travel a given distance upstream as it does downstream.

Answer: A) Upstream speed is 10 km/h, B) Downstream speed is 20 km/h, C) Twice as long upstream

### Question types: Number entry

###### Difficulty level: hard

Question 1:

A car travels from Town A to Town B at an average speed of 50 km/h and then returns to Town A at an average speed of 70 km/h. The round trip takes 6 hours. What is the distance between Town A and Town B in kilometers?

Question 2:

A train leaves Station A and travels to Station B at a constant speed of 80 km/h. It then leaves Station B and travels to Station C at a constant speed of 100 km/h. The entire journey from Station A to Station C takes 4 hours. If the distance between Station A and Station B is twice the distance between Station B and Station C, what is the distance between Station A and Station C in kilometers?

Question 3: Two runners start at the same point on a circular track with a circumference of 1,200 meters. Runner A runs at a constant speed of 15 km/h, and Runner B runs at a constant speed of 12 km/h. After how many minutes will Runner A lap Runner B for the first time?

Question 4: A cyclist travels 30 km from Town A to Town B, then 40 km from Town B to Town C. The average speed for the entire trip is 25 km/h. If the cyclist traveled at an average speed of 20 km/h from Town A to Town B, what was the average speed in km/h from Town B to Town C?

### Question types: Number entry

###### Difficulty level: Challenging

Question 1:

A car travels from City A to City B at a constant speed of 60 km/h and takes 2 hours longer than it would have taken if the car had traveled at a constant speed of 90 km/h. What is the distance between City A and City B in kilometers?

Question 2: Two cyclists start from opposite ends of a 150 km route at the same time. Cyclist A has a constant speed of 25 km/h, while Cyclist B has a constant speed of 35 km/h. At what distance will the two cyclists meet from Cyclist A’s starting point?