Lesson 4.2: Rate, Distance, and Work Problems
In this lesson, we will explore the concepts of rate, distance, and work problems in quantitative reasoning. These topics are essential for solving various problems in the GMAT, especially since many of the quantitative questions you will encounter are structured as word problems. By the end of this lesson, you should be able to understand and apply the rate-time-distance relationship and the work-rate relationship effectively.
Learning Objectives
- Understand the rate-time-distance relationship and the work-rate relationship.
- Tackle combined-rate and multiple-worker scenarios.
- Set up tables to organize rate problems.
- Solve distance and work problems involving single and combined rates.
- Organize rate information to avoid setup errors.
1. The Rate-Time-Distance Relationship
The rate-time-distance relationship is fundamental in solving problems that involve motion. The primary formula that describes this relationship is:
$$
$\text{Distance} = \text{Rate} \times \text{Time}$
$$
1.1 Understanding the Components
- Distance (D): The total length of space traveled, typically measured in units like miles, kilometers, or meters.
- Rate (R): The speed at which the object is traveling, often in miles per hour (mph) or kilometers per hour (km/h).
- Time (T): The duration for which the object has been traveling, generally measured in hours, minutes, or seconds.
1.2 Rearranging the Formula
From the main formula, we can derive other useful equations by rearranging the terms:
- To find the rate:
$$
$ R = \frac{D}{T}$
$$
- To find the time:
$$
$ T = \frac{D}{R}$
$$
1.3 Worked Example
Example 1: A car travels 150 miles at a speed of 50 miles per hour. How long does the journey take?
Solution:
We can use the formula for time:
$$
$T = \frac{D}{R}$
$$
Substituting in the values:
$$
T = $\frac{150 \text{ miles}}{50 \text{ mph}}$ = $3 \text{ hours}$
$$
Thus, the journey takes 3 hours.
2. Work-Rate Problems
Work-rate problems involve multiple individuals or machines working together to complete a task within a certain time frame. The primary relationship we use is:
$$
$\text{Work Done} = \text{Rate} \times \text{Time}$
$$
2.1 Understanding Work Rate
- Work Done: The total amount of work completed, often measured in tasks, units, or jobs.
- Rate: This is how quickly the work is being done, particularly when dealing with multiple workers.
- Time: The amount of time spent on the work.
2.2 Combined Work Rate
If multiple workers are involved, their rates are additive. So, if Worker A can complete a job in 4 hours, and Worker B can complete it in 6 hours, their rates are:
- Worker A’s rate: $$R_A = \frac{1 \text{ job}}{4 \text{ hours}} = \frac{1}{4} \text{ jobs/hour}$$
- Worker B’s rate: $$R_B = \frac{1 \text{ job}}{6 \text{ hours}} = \frac{1}{6} \text{ jobs/hour}$$
Their combined rate is:
$$
R_{A+B} = R_A + R_B = $\frac{1}{4}$ + $\frac{1}{6}$ = $\frac{3}{12}$ + $\frac{2}{12}$ = $\frac{5}{12}$ $\text{ jobs/hour}$
$$
2.3 Worked Example
Example 2: If Worker A can complete a job in 5 hours and Worker B in 10 hours, how long will it take them to complete the job if they work together?
Solution:
First, we determine their rates:
- Worker A’s rate: $$R_A = \frac{1 \text{ job}}{5 \text{ hours}} = \frac{1}{5} \text{ jobs/hour}$$
- Worker B’s rate: $$R_B = \frac{1 \text{ job}}{10 \text{ hours}} = \frac{1}{10} \text{ jobs/hour}$$
Thus, the combined rate is:
$$
R_{A+B} = $\frac{1}{5}$ + $\frac{1}{10}$ = $\frac{2}{10}$ + $\frac{1}{10}$ = $\frac{3}{10}$ $\text{ jobs/hour}$
$$
Now, to find the total time to complete 1 job:
$$
T = $\frac{1 \text{ job}}{R_{A+B}}$ = $\frac{1 \text{ job}}{\frac{3}{10} \text{ jobs/hour}}$ = $\frac{10}{3}$ $\text{ hours}$ $\approx 3$.$33 \text{ hours}$
$$
So, together, they take approximately 3.33 hours to finish the job.
3. Organizing Rate Problems with Tables
Organizing information in a table can be extremely useful in avoiding common setup errors and ensuring you capture all relevant data. Consider creating a table with columns for each participant, their rate, time, and work done. This setup can help visualize the situation clearly.
3.1 Example Setup
Example 3: A train leaves Station A at 3 PM, traveling at 60 mph, while another train leaves Station B at 4 PM, traveling at 75 mph. When will they meet?
Step 1: Create a Table
| Train | Departure Time | Rate (mph) | Time (hours) | Distance (miles) |
|---|---|---|---|---|
| Train A | 3 PM | 60 | $T_A$ | $D_A = 60T_A$ |
| Train B | 4 PM | 75 | $T_B$ | $D_B = 75T_B$ |
Step 2: Establish Relationships
Since Train A departs an hour earlier, we can express the times as follows:
- Train A’s time: $T_A = T$ hours
- Train B’s time: $T_B = T - 1$ hours
Step 3: Write the Distance Equations
Both trains travel towards each other until they meet, which means:
$$
$D_A + D_B = \text{Total Distance}$
$$
Substituting in our expressions:
$$
60T + 75(T - 1) = 60T + 75T - 75
$$
Step 4: Solve for T
This leads to:
$$
$135T - 75 = \text{Total Distance}$
$$
You continue this to find the exact meeting time based on the distances involved.
Conclusion
In conclusion, mastering rate, distance, and work problems is crucial for the GMAT and other standardized tests. You'll find that practice is key to gaining confidence in recognizing setups and solutions. Make sure to set your problems correctly to streamline the process and reduce errors.
Study Notes
- The fundamental relationship: Distance = Rate × Time.
- In work problems, rates are additive when multiple workers are involved.
- Use tables to clarify rates, times, and distances for complex problems.
- Always check units to ensure consistency and correctness in calculations.
- Combining rates encourages a systematic and organized approach to solving problems.