Table of Contents
Key Takeaways:
• Time and work problems GMAT use the formula: Work = Rate × Time
• Add individual rates to solve problems with people working together
• Use 1/x to find the rate if one person finishes a task in x hours
• LCM helps simplify tough problems with fractions or tricky values
• Don’t add time directly; always add rates first to find the total time
Time and work problems frequently appear in the GMAT Quant section. These questions usually ask how long it takes for people or machines to finish a task, either alone or by working together. The good thing is, that they all follow one basic rule: work equals rate multiplied by time. Once you understand that, the rest becomes much easier.
You might see questions like two people painting a wall or a machine making boxes. To solve them, you simply need to determine the speed of each person or machine and then use that information to find the answer. With a bit of practice, you’ll spot patterns and solve these problems quickly on test day.
How to Solve GMAT Work Problems
Time and work problems are common in the GMAT Quant section and focus on finding out how long it will take for one or more people (or machines) to complete a task when working at certain rates. These problems can be solved using simple formulas and methods.
One of the main formulas used in time and work problems is:
Work Done = Rate × Time
This formula helps in calculating either the time required to complete a task or the rate of work if the time and amount of work are known.
Example:
If person A can complete a task in 5 hours and person B can do it in 10 hours, their combined work rate is calculated as:
- A's rate = 1/5 (one task in 5 hours)
- B's rate = 1/10 (one task in 10 hours)
Their combined rate = 15+110=310\{1}{5} + \{1}{10} = \{3}{10}51+101=103
So together, they will complete 310\{3}{10}103 of the task per hour, and the total time required to complete the work is:
Total Time = 1 / Combined Rate = 10/3 hours or 3.33 hours.
What is the Work Rate Formula?
The Work Rate Formula is a key concept in solving time and work problems on the GMAT. It calculates how much work is done by an individual or a group of individuals within a certain time frame.
The formula can be written as:
Work Rate = Work Done / Time Taken
This formula can be applied to both individual work rates and combined work rates when more than one person or machine is working together. Let’s break it down with another example:
Example:
Suppose Machine A can produce 200 units in 4 hours, and Machine B can produce 200 units in 8 hours. Their individual work rates would be:
- Machine A’s rate = 200 units / 4 hours = 50 units per hour
- Machine B’s rate = 200 units / 8 hours = 25 units per hour
If they work together, their combined rate would be:
Combined Rate = 50 + 25 = 75 units per hour
So, working together, they can produce 200 units in:
Time = Work Done / Combined Rate = 200 / 75 = 2.67 hours
| Individual/Machine | Work Done (Units) | Time Taken (Hours) | Work Rate (Units per Hour) |
|---|---|---|---|
| Machine A | 200 | 4 | 50 |
| Machine B | 200 | 8 | 25 |
| Combined Rate | 200 | 2.67 | 75 |
Please refer to GMAT rate problems for a detailed analysis of time and work problems GMAT
Time and Work Concept in GMAT
The time and work concept revolves around understanding how different work rates come together to complete a task within a specific time frame. In GMAT problems, you're often asked to calculate the total time required to finish a task when multiple people or machines work together or individually.
Key Principles:
Work Rate: This represents how much work an individual or a machine can do in a given time period. It's typically expressed as "work done per unit of time."
Combined Work Rate: When multiple individuals work together, their work rates add up. This allows you to determine the total time required to complete a job.
Inverse Relation of Time and Work: If the number of people working increases, the time taken decreases. Conversely, fewer workers means more time will be needed.
Formula Breakdown:
Individual Rate = 1TimeTaken\{1}{Time Taken}TimeTaken1
Combined Rate = Sum of Individual Rates
Total Time = 1CombinedRate\{1}{Combined Rate}CombinedRate1
Example:
Consider a scenario where person A can complete a task in 4 hours, and person B can do the same task in 6 hours. Their individual work rates would be:
A's rate = 14\{1}{4}41 (task per hour)
B's rate = 16\{1}{6}61 (task per hour)
Their combined work rate is:
- 14+16=312+212=512\{1}{4} + \{1}{6} = \{3}{12} + \{2}{12} = \{5}{12}41+61=123+122=125
- This means together, they complete 512\{5}{12}125 of the work in one hour. The total time taken to finish the task would be:
- Total Time = 1512\{1}{\{5}{12}}1251 = 2.4 hours.
This example demonstrates how two people working together can complete a task faster than working individually.
Real-life Application:
Many time and work problems on the GMAT involve people working together or at different rates. For instance, you may be asked to calculate how long it takes for two workers to paint a house, fill a tank, or complete a set of tasks when their working speeds differ.
GMAT Quant 800 Level Problem for Time and Work
The 800-level questions in the GMAT Quant section are the most challenging and are designed to test your deep understanding of concepts and your ability to apply them under time pressure. Time and work problems at this level often involve multiple layers of reasoning, complex calculations, and the combination of different problem-solving techniques.
Key Elements of 800-Level Time and Work Problems:
Complex Rates: These problems typically involve more than two workers or machines, each with a different work rate. The rates may involve fractions or require conversions between units of time (e.g., hours to minutes).
Variable Work Rates: In some cases, the rates of workers or machines change over time, requiring dynamic calculations.
Partial Work Problems: These problems may ask you to calculate the time taken to complete a fraction of the task or involve workers starting and stopping at different times.
Example of an 800-Level Time and Work Problem:
Problem:
Machines A, B, and C can complete a task individually in 12, 15, and 20 hours, respectively. Machine A starts working on the task and works alone for 3 hours. After 3 hours, Machine B joins A, and they work together for the next 4 hours. Finally, Machine C joins A and B, and all three machines complete the remaining task. How long did it take for the entire task to be completed?
Solution:
Step 1: Calculate the work done by Machine A in 3 hours
Machine A's rate = 112\frac{1}{12}121 of the task per hour.
Work done by A in 3 hours = 3×112=312=143 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4}3×121=123=41.
After 3 hours, 14\frac{1}{4}41 of the task is completed, and 34\frac{3}{4}43 remains.
Step 2: Calculate the work done by A and B together in the next 4 hours
Machine B’s rate = 115\frac{1}{15}151 of the task per hour.
Combined rate of A and B = 112+115\frac{1}{12} + \frac{1}{15}121+151.
To add these, find the least common denominator (LCD) of 12 and 15, which is 60:
112=560,115=460\frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}121=605,151=604Combined rate = 560+460=960=320\frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20}605+604=609=203 of the task per hour.
Work done by A and B in 4 hours = 4×320=1220=354 \times \frac{3}{20} = \frac{12}{20} = \frac{3}{5}4×203=2012=53.
After 4 hours, an additional 35\frac{3}{5}53 of the task is completed. At this point, 14+35\frac{1}{4} + \frac{3}{5}41+53 of the task is done.
Convert both fractions to the same denominator (20):
14=520,35=1220\frac{1}{4} = \frac{5}{20}, \quad \frac{3}{5} = \frac{12}{20}41=205,53=2012Total work done so far = 520+1220=1720\frac{5}{20} + \frac{12}{20} = \frac{17}{20}205+2012=2017.
Only 320\frac{3}{20}203 of the task remains.
Step 3: Calculate the time taken by A, B, and C to finish the remaining work
Machine C’s rate = 120\frac{1}{20}201 of the task per hour.
Combined rate of A, B, and C = 112+115+120\frac{1}{12} + \frac{1}{15} + \frac{1}{20}121+151+201.
Find the least common denominator (LCD) of 12, 15, and 20, which is 60:
112=560,115=460,120=360\frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60}121=605,151=604,201=603Combined rate = 560+460+360=1260=15\frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5}605+604+603=6012=51 of the task per hour.
Time to complete the remaining 320\frac{3}{20}203 of the task:
Time=32015=320×51=1520=0.75 hours.\text{Time} = \frac{\frac{3}{20}}{\frac{1}{5}} = \frac{3}{20} \times \frac{5}{1} = \frac{15}{20} = 0.75 \text{ hours}.Time=51203=203×15=2015=0.75 hours.Therefore, the total time taken is:
3 hours+4 hours+0.75 hours=7.75 hours.3 \text{ hours} + 4 \text{ hours} + 0.75 \text{ hours} = 7.75 \text{ hours}.3 hours+4 hours+0.75 hours=7.75 hours.
Answer: The entire task was completed in 7.75 hours.
Please refer GMAT Work Rate Problems for detailed analysis of Time and Work problems GMAT
GMAT vs GRE
Efficiency Method to Solve Time and Work
The efficiency method is another useful strategy to tackle GMAT time and work problems. In this method, efficiency refers to the rate at which work is completed, expressed as a percentage or a ratio. The higher the efficiency, the faster the task will be completed.
Formula:
- Efficiency of A = WorkDoneTimeTaken\{Work Done}{Time Taken}TimeTakenWorkDone
- Combined Efficiency = Sum of individual efficiencies
- Total Time = TotalWorkCombinedEfficiency\{Total Work}{Combined Efficiency}CombinedEfficiencyTotalWork
Example:
Suppose Machine A and Machine B have different efficiencies. Machine A completes 60% of a task in 3 hours, while Machine B completes 40% of the task in the same time. Their combined efficiency would be:
- Machine A's efficiency = 603=20%\{60}{3} = 20\%360=20% per hour
- Machine B's efficiency = 403=13.33%\{40}{3} = 13.33\%340=13.33% per hour
The total efficiency is:
- Total Efficiency = 20% + 13.33% = 33.33% per hour.
Now, to calculate the total time taken to complete the full task:
Total Time = ( \{100%}{33.33%} = 3 hours.
Thus, the machines working together would complete the task in 3 hours.
Practical Use of Efficiency:
The efficiency method is particularly helpful when dealing with complex work rate problems on the GMAT, where different workers or machines operate at varied speeds. This approach helps streamline the calculations and avoid unnecessary complications.
Example Table: Efficiency Method (HTML)
| Machine/Worker | Efficiency (% per hour) | Time Taken (hours) |
|---|---|---|
| Machine A | 20% | 3 |
| Machine B | 13.33% | 3 |
| Combined Efficiency | 33.33% | 3 |
Tricks to Solve Work Problems Quickly
When solving time and work problems on the GMAT, speed and accuracy are crucial. There are several tricks and shortcuts you can use to solve these problems more efficiently. Understanding these methods can help save time and improve accuracy during the exam.
Key Tricks:
LCM Method:
The Least Common Multiple (LCM) method simplifies work problems by reducing the complexity of dealing with fractions. Instead of calculating complex rates, you find the LCM of the times involved and use that to calculate the total work done.
Example: Suppose Worker A can complete a task in 4 hours, and Worker B can complete the same task in 6 hours. The LCM of 4 and 6 is 12. This means the total work is 12 units (assuming 1 unit of work is completed in the least amount of time, 1 hour).
- Worker A’s rate = 124=3\{12}{4} = 3412=3 units per hour
- Worker B’s rate = 126=2\{12}{6} = 2612=2 units per hour
Together, A and B’s combined rate is 3+2=53 + 2 = 53+2=5 units per hour. Therefore, they will take:
Total Time = 125\{12}{5}512 hours = 2.4 hours to complete the work together.
Efficiency Formula:
Another trick is using efficiency, where you calculate how much work is done per unit of time by each worker. If you know the efficiency, you can find the total time or work done by simply dividing or multiplying values.
Example: If Machine X completes 60% of a job in 4 hours, its efficiency is 604=15%\{60}{4} = 15\%460=15% per hour. This means Machine X does 15% of the work every hour.
Using Proportions:
When dealing with multiple people or machines working together, proportions can help solve problems faster. The total time taken is inversely proportional to the combined efficiency.
Example: If Machine A works at 50% efficiency and Machine B works at 30%, their combined efficiency is 80%. This means they can finish the job in 10080\{100}{80}80100 hours or 1.25 hours.
| Mistake | Description | Solution |
|---|---|---|
| Ignoring Combined Work Rate | Forgetting to add individual rates | Always sum individual rates when working together |
| Misunderstanding Units | Mixing up units like hours and minutes | Convert all units to the same measure before solving |
| Skipping Steps | Not using LCM or simplifying fractions | Break the problem into small, manageable steps |
| Incorrect Formula Application | Misusing the work formula | Remember Work = Rate × Time |
Example Problems with Solutions
In GMAT time and work problems, it is essential to practice using different strategies to solve them quickly and accurately. Below are some examples to help you understand the types of questions that might appear and how to solve them step by step.
Example 1: Two Workers with Different Speeds
Let’s say Person A can complete a task in 6 hours, while Person B can complete the same task in 8 hours. How long will it take for both of them to finish the task if they work together?
Solution:
Calculate each person’s rate of work:
- A’s rate = 16\{1}{6}61 of the task per hour.
- B’s rate = 18\{1}{8}81 of the task per hour.
Add their rates together to find the combined rate:
Combined rate = 16+18\{1}{6} + \{1}{8}61+81.
Find the least common denominator (LCD), which is 24, and add the fractions: 16=424,18=324\{1}{6} = \{4}{24}, \quad \{1}{8} = \{3}{24}61=244,81=243.
Combined rate = 424+324=724\{4}{24} + \{3}{24} = \{7}{24}244+243=247 of the task per hour.
Calculate the total time taken:
- Total time = 1724\{1}{\{7}{24}}2471.
- This equals 247≈3.43\{24}{7} \approx 3.43724≈3.43 hours.
Thus, A and B together will complete the task in about 3.43 hours.
Example 2: Machine Work Rate Problem
A factory has two machines, Machine X and Machine Y. Machine X can complete a job in 10 hours, and Machine Y can complete the same job in 15 hours. If both machines work together, how long will it take to complete the job?
Solution:
Calculate each machine’s rate of work:
- X’s rate = 110\{1}{10}101 of the job per hour.
- Y’s rate = 115\{1}{15}151 of the job per hour.
Add their rates to get the combined rate: 110+115\{1}{10} + \{1}{15}101+151.
Find the LCD (30), and rewrite the fractions: 110=330,115=230\{1}{10} = \{3}{30}, \quad \{1}{15} = \{2}{30}101=303,151=302.
Combined rate = 330+230=530=16\{3}{30} + \{2}{30} = \{5}{30} = \{1}{6}303+302=305=61.
Calculate the total time:
Total time = 116=6\{1}{\{1}{6}} = 6611=6 hours.
Thus, both machines working together will complete the job in 6 hours.
Commonly Asked GMAT Work Questions
Certain types of time and work problems frequently appear in GMAT exams. Being familiar with these common question types can help you prepare more efficiently. Here are a few examples of commonly asked questions:
Question Type 1: Work Done by Multiple Workers
If multiple workers are involved in completing a task, the question will ask for the total time required if they work together or how much work they can complete in a given time. These questions typically involve adding the work rates of each worker.
Example: Person A can complete a job in 4 hours, and Person B can complete the same job in 5 hours. How long will it take both of them to complete the task together?
Answer:
- A’s rate = 14\{1}{4}41, B’s rate = 15\{1}{5}51.
- Combined rate = 14+15=520+420=920\{1}{4} + \{1}{5} = \5}{20} + \{4}{20} = \{9}{20}41+51=205+204=209 of the task per hour.
- Time taken = 1920=209≈2.22\{1}{\{9}{20}} = \{20}{9} \approx 2.222091=920≈2.22 hours.
Question Type 2: Machine Work Rates
These questions involve machines or pumps working at different speeds. The goal is to calculate how long it will take for the machines to finish a task if they work together.
Example: Machine A can complete a task in 12 hours, and Machine B can complete the same task in 18 hours. If both machines work together, how long will it take to finish the task?
Answer:
- A’s rate = 112\{1}{12}121, B’s rate = 118\{1}{18}181.
- Combined rate = 112+118=336+236=536\{1}{12} + \{1}{18} = \{3}{36} + \{2}{36} = \{5}{36}121+181=363+362=365.
- Time taken = 1536=365=7.2\{1}{\{5}{36}} = \{36}{5} = 7.23651=536=7.2 hours.
Question Type 3: Partial Work Problems
Some questions ask about situations where one person or machine starts the work, and another finishes it, or where one works for a certain time before stopping.
Example: Person A can finish a task in 6 hours. Person B starts the task and works for 2 hours, after which A finishes the rest of the task. How long will it take to complete the task?
Answer:
- Let’s say B works for 2 hours at a rate of 16\{1}{6}61 per hour, so B completes 2×16=132 \times \{1}{6} = \{1}{3}2×61=31 of the task.
- A must complete 1−13=231 - \{1}{3} = \{2}{3}1−31=32 of the task. A’s rate is 16\{1}{6}61, so the time taken by A to finish the rest is: 23×6=4 hours.\{2}{3} \times 6 = 4 \text{ hours}.32×6=4 hours.
- Total time taken = 2 (by B) + 4 (by A) = 6 hours.
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Conclusion
Mastering time and work problems in the GMAT Quant section requires a solid understanding of basic formulas and how to apply them efficiently. Whether you are calculating individual or combined work rates, breaking down the problem step-by-step is key to solving it quickly and accurately. By practicing these types of questions, you can build confidence and reduce errors during the exam.
