6.4 Rates, Work, Statistics & Word Problems
Key Takeaways
- For distance problems, use the formula Distance = Rate × Time (D = RT). Average speed is Total Distance / Total Time, not the average of the speeds.
- In combined work problems, add the rates, not the times: 1/Total Time = 1/Time1 + 1/Time2.
- Understand the difference between mean (average), median (middle value), and mode (most frequent value).
- Use the Overlapping Sets formula: Total = Group A + Group B - Both + Neither.
Rates, Work, Statistics & Word Problems
Many of the most daunting word problems on the EA fall into a few predictable categories: rates, work, statistics, and sets. Recognizing the category allows you to immediately apply the correct framework, turning a paragraph of text into a simple solvable equation.
Distance, Rate, and Time
The fundamental formula for motion problems is: Distance = Rate × Time (D = RT)
When dealing with multiple trips or segments, use a D-R-T chart to organize the information.
| Trip Segment | Rate | Time | Distance (R × T) |
|---|---|---|---|
| Going | r1 | t1 | d1 |
| Returning | r2 | t2 | d2 |
| Total | --- | t1 + t2 | d1 + d2 |
Crucial Rule for Average Speed: Average Speed is NEVER the simple average of two speeds. It is strictly: Average Speed = Total Distance / Total Time
Example: You drive to a city at 40 mph and return along the same route at 60 mph. What is your average speed? Let the distance one-way be 120 miles (a convenient multiple). Time there = 120 / 40 = 3 hours. Time back = 120 / 60 = 2 hours. Total Distance = 240 miles. Total Time = 5 hours. Average Speed = 240 / 5 = 48 mph. (Notice it is not 50 mph!).
Combined Work Rates
Work problems are just rate problems where "Distance" is replaced by "Work" (usually 1 completed job). Work = Rate × Time.
When multiple entities work together, you add their rates, not their times. If Machine A takes 4 hours to do a job (rate = 1/4 job/hour) and Machine B takes 6 hours to do the job (rate = 1/6 job/hour), their combined rate is: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 job per hour.
The time to complete the job together is the reciprocal of the combined rate: 12/5 hours, or 2.4 hours.
The general formula is: 1/T_total = 1/T_1 + 1/T_2 + ...
Descriptive Statistics
The EA expects you to know the fundamental measures of central tendency and dispersion:
- Mean (Average): The sum of the terms divided by the number of terms. Formula: Sum = Mean × Number of Terms. (This rearranged version is often more useful).
- Median: The middle value when the numbers are arranged in ascending order. If there is an even number of terms, the median is the average of the two middle numbers.
- Mode: The number that appears most frequently in a set. A set can have multiple modes or no mode.
- Range: The difference between the highest and lowest values in a set.
- Standard Deviation (SD): A measure of how spread out the numbers are from the mean. A small SD means the numbers are clustered tightly around the mean; a large SD means they are widely dispersed. You rarely have to calculate the exact SD; you just need to understand the concept of variance. If you add a constant to every term in a set, the SD does not change. If you multiply every term by a constant, the SD is multiplied by the absolute value of that constant.
Weighted Averages
When calculating an average where different components have different "weights" or quantities, you cannot use a simple average.
Example: A class has 10 boys with an average score of 80, and 20 girls with an average score of 92. What is the class average? Total boy points = 10 × 80 = 800. Total girl points = 20 × 92 = 1840. Total points = 2640. Total students = 30. Weighted Average = 2640 / 30 = 88. Notice the average (88) is closer to the girls' average (92) because there are more girls.
Overlapping Sets (Venn Diagrams)
For problems involving two groups where some members belong to both, use the Overlapping Sets formula:
Total = Group A + Group B - Both + Neither
Why subtract "Both"? Because when you add Group A and Group B, you are counting the people in both groups twice. Subtracting "Both" once corrects the overcounting.
Example: In a group of 50 students, 30 take French, 25 take Spanish, and 10 take neither. How many take both? Total = French + Spanish - Both + Neither 50 = 30 + 25 - Both + 10 50 = 65 - Both Both = 15 students.
Basic Counting and Probability
- Fundamental Counting Principle: If there are m ways to do one thing and n ways to do another, there are m × n ways to do both.
- Probability: The number of desired outcomes divided by the total number of possible outcomes. Probability is always a fraction between 0 (impossible) and 1 (certain).
- P(A or B) for mutually exclusive events: P(A) + P(B)
- P(A and B) for independent events: P(A) × P(B)
By systematizing your approach to these word problems with charts and standard formulas, you can navigate them quickly without getting lost in the prose.
A train travels from Station A to Station B at a speed of 60 miles per hour, and returns from B to A at a speed of 40 miles per hour. What is the train's average speed for the entire round trip?
Pump A can fill a tank in 3 hours. Pump B can fill the same tank in 6 hours. If both pumps work together, how long will it take to fill the tank?
In a company of 100 employees, 60 employees have a gym membership, 45 employees have a transit pass, and 20 employees have neither. How many employees have both a gym membership and a transit pass?