To solve time and work problems effectively for the GRE, you must grasp the core relationship between work, rate, and time. These questions often assess your ability to calculate the combined effort of individuals or groups completing tasks. This guide offers a clear, concise approach to tackle these problems, providing both the theory and practical examples to enhance your problem-solving skills.
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Welcome to a focused guide on time and work problems, a common feature of the GRE Quantitative Reasoning section. These problems test your ability to calculate the rate at which individuals or groups complete tasks. You'll learn to solve time and work questions efficiently and accurately.
Understanding the Fundamentals
Time and work problems typically involve individuals or groups working together or separately to complete a task. The core concept is to understand the relationship between the amount of work, the rate of work, and the time taken. The rate of work is often expressed as the amount of work done per unit of time (e.g., tasks per hour).
Basic Principles and Formulas
The fundamental formula for solving time and work problems is: Work = Rate × Time. This formula can be rearranged to solve for any of the variables: Rate = Work / Time and Time = Work / Rate. Understanding these relationships is key. The work can be represented as a whole, such as the completion of a task. The rate is the portion of work completed in a given time.
When individuals work together, their rates are added. For instance, if person A can complete a task in 4 hours and person B can complete the same task in 6 hours, their combined rate can be calculated. Consider a scenario where two people, A and B, are working on a project. A can finish it in 20 days and B in 30 days. The combined rate of work is determined by calculating the individual rates and summing them. Person A's rate is 1/20 of the work per day, and person B's rate is 1/30 of the work per day. When working together, their combined rate is ##1/20 + 1/30 = 1/12##. This means they will finish the work in 12 days.
In problems where the work is not explicitly stated, you can assume the total work to be a unit (1). This simplifies calculations, especially when dealing with fractions of work. For example, if a person completes a fraction of a task, the remaining work can be calculated by subtracting the fraction completed from 1. Another example involves three workers, X, Y, and Z, who can complete a task individually in 10, 12, and 15 days, respectively. To find how long they take to complete the work together, we calculate their combined rate by adding their individual rates: ##1/10 + 1/12 + 1/15 = 1/4##. This indicates that, working together, they can finish the task in 4 days.
Solving Time and Work Problems: A Step-by-Step Approach
To effectively solve time and work problems, a systematic approach is essential. Begin by identifying the key information: the total work (if given), the rates of individuals or groups, and the time taken. Next, convert the given information into a usable format, often by calculating rates or expressing the work as a unit. Finally, apply the formulas to find the unknown quantity, such as the time required or the rate of work.
Example Problem 1
Problem: If John can paint a house in 4 hours and Mary can paint the same house in 6 hours, how long will it take them to paint the house if they work together? Solution: John's rate = 1/4 house/hour, Mary's rate = 1/6 house/hour. Combined rate = 1/4 + 1/6 = 5/12 house/hour. Time = Work / Rate = 1 / (5/12) = 12/5 hours or 2.4 hours. Thus, it will take them 2.4 hours to paint the house together.
Step-by-step solution: Calculate individual rates: John's rate is 1/4, and Mary's is 1/6. Sum the rates: 1/4 + 1/6 = 5/12. Calculate time using the combined rate: Time = 1 / (5/12) = 2.4 hours. The combined work rate is ##1/4 + 1/6##, which simplifies to ##5/12##. Hence, if they work together, they finish the work in ##1 / (5/12) = 2.4## hours.
In this problem, the work is the house, and the rates are expressed in terms of the fraction of the house painted per hour. When they work together, we add their rates, and the combined rate is used to calculate the time. For example, if two pipes can fill a tank separately in 20 and 30 minutes, the combined filling rate is found by adding their individual filling rates, and the time it takes to fill the tank is calculated using the combined rate.
More Practice Problems
To enhance your skills, try more practice problems. These examples will solidify your understanding of the concepts. Remember to break down each problem into its components: rate, time, and work. Practice is key to mastering these types of questions on the GRE.
Example 2
Problem: A can complete a work in 20 days and B can complete the same work in 30 days. In how many days can they complete the work together? Solution: A's one day work = 1/20, B's one day work = 1/30. (A+B)'s one day work = 1/20 + 1/30 = 1/12. Therefore, A and B together can complete the work in 12 days.
A can finish a task in 20 days, so A's daily work rate is ##1/20##. B can finish the same task in 30 days, making B's daily work rate ##1/30##. The combined rate is ##1/20 + 1/30 = 1/12##, therefore they finish the work in 12 days. If A can finish a work in 10 days and B can finish the same work in 15 days, then the combined time is found by first calculating their rates as ##1/10## and ##1/15##, respectively, and then summing them to find their combined work rate of ##1/6##. Hence, they complete the work in 6 days.
If John takes 4 hours to paint a house, his work rate is ##1/4## per hour. If Mary takes 6 hours, her work rate is ##1/6## per hour. Their combined rate is the sum of their individual rates, which is ##1/4 + 1/6##. The combined rate is used to find the total time taken to complete the work together, which is 2.4 hours. If A and B can complete a task in 20 and 30 days, respectively, then their combined work rate is found by first calculating their individual rates, ##1/20## and ##1/30##, and then summing them to get ##1/12##. Hence, they complete the work in 12 days.
Key Takeaways
Mastering time and work problems involves understanding the relationship between work, rate, and time. By systematically approaching each problem, calculating rates, and applying the appropriate formulas, you can improve your accuracy and efficiency. Remember to practice these types of problems to build confidence for the GRE.
| Problem | Solution | Answer |
|---|---|---|
| If A can do a piece of work in 20 days and B can do it in 30 days, how long will they take to complete the work together? | A's 1-day work = 1/20, B's 1-day work = 1/30. (A+B)'s 1-day work = 1/20 + 1/30 = 1/12 | 12 days |
| A can finish a work in 10 days and B can finish the same work in 15 days. How long will they take to complete the work together? | A's 1-day work = 1/10, B's 1-day work = 1/15. (A+B)'s 1-day work = 1/10 + 1/15 = 1/6 | 6 days |
| John can paint a house in 4 hours and Mary can paint the same house in 6 hours. How long will it take them to paint the house if they work together? | John's rate = 1/4, Mary's rate = 1/6. Combined rate = 1/4 + 1/6 = 5/12. Time = 1 / (5/12) | 2.4 hours |
| Pipe A can fill a tank in 20 minutes, and Pipe B can fill it in 30 minutes. How long will it take to fill the tank if both pipes are open? | Pipe A's rate = 1/20, Pipe B's rate = 1/30. Combined rate = 1/20 + 1/30 | 12 minutes |
| If A and B together can complete a work in 12 days, and A alone can complete it in 20 days, in how many days can B alone complete it? | (A+B)'s 1-day work = 1/12, A's 1-day work = 1/20. B's 1-day work = 1/12 - 1/20 | 30 days |
| A can do a work in 10 days. B can do the same work in 15 days. They work together for 4 days. How much work is left? | A's 1-day work = 1/10, B's 1-day work = 1/15. (A+B)'s 1-day work = 1/6. Work done in 4 days = 4/6 | 1/3 |
| A can do a piece of work in 10 days, and B can do the same work in 15 days. If they work together, how much work will they complete in 2 days? | A's 1-day work = 1/10, B's 1-day work = 1/15. (A+B)'s 1-day work = 1/6. Work done in 2 days = 2/6 | 1/3 |
| A and B can complete a work in 10 days. A alone can complete the work in 15 days. In how many days can B alone complete the work? | (A+B)'s 1-day work = 1/10, A's 1-day work = 1/15. B's 1-day work = 1/10 - 1/15 | 30 days |
| A can complete a work in 12 days. B is 60% more efficient than A. How many days does B take to do the same work? | B's efficiency = 1.6 times A's. If A takes 12 days, B takes 12/1.6 | 7.5 days |
| A can do a piece of work in 24 days. B is 20% more efficient than A. How many days does B take to complete the same work? | B's efficiency = 1.2 times A's. If A takes 24 days, B takes 24/1.2 | 20 days |
| A can do a work in 18 days, and B can do the same work in 24 days. They work together for 8 days, and then A leaves. How many days will B take to complete the remaining work? | A's 1-day work = 1/18, B's 1-day work = 1/24. (A+B)'s 1-day work = 1/18 + 1/24. Work done in 8 days = 8*(1/18 + 1/24) | 6 days |
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