Of course. Here is a complete, in-depth article about a GRE practice question of the day, crafted to be both educational and SEO-friendly.
Mastering GRE Algebra: A Practice Question on Combined Work Rates
Conquer the GRE with our daily practice question! Today, we’re diving into a classic algebra problem involving combined work rates, a frequent topic on the GRE Quantitative Reasoning section. This type of question tests your ability to set up and solve equations based on how different entities work together. By breaking down the problem systematically, you can build the confidence to tackle even the most complex rate questions on test day Surprisingly effective..
Problem: Working alone at their respective constant rates, Machine A would take 12 hours to complete a certain job. Machine B, working alone at its constant rate, would take 18 hours to complete the same job. If both machines work together at their respective constant rates, how many hours would it take them to complete the job?
This question is a staple because it moves beyond simple arithmetic and requires a fundamental conceptual shift. Let’s dissect it.
The Core Concept: The Work Rate Formula
The secret to solving combined work problems lies in understanding that when things work together, their rates add up. The fundamental formula is:
Work = Rate × Time
From this, we can derive the rate: Rate = Work / Time
The "Work" in this context is often the completion of one entire job. So, if Machine A takes 12 hours to complete 1 job, its work rate is 1/12 of the job per hour. Similarly, Machine B's rate is 1/18 of the job per hour That alone is useful..
Step-by-Step Solution
Let’s apply this concept to our problem The details matter here..
Step 1: Define the Variables and Rates Let the entire job be represented by 1 (one complete job) Simple, but easy to overlook..
- Rate of Machine A (R_A) = 1 job / 12 hours = 1/12 job per hour.
- Rate of Machine B (R_B) = 1 job / 18 hours = 1/18 job per hour.
Step 2: Determine the Combined Rate When working together, their rates are additive. This is the most critical step.
- Combined Rate (R_C) = R_A + R_B
- R_C = (1/12) + (1/18)
To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 18 is 36.
- 1/12 = 3/36
- 1/18 = 2/36
- That's why, R_C = 3/36 + 2/36 = 5/36 job per hour.
So in practice, together, the machines complete 5/36 of the job in one hour.
Step 3: Calculate the Time Required Now, we use the original formula, rearranged to solve for Time: Time = Work / Rate
Here, the Work is 1 (the entire job), and the Rate is the Combined Rate (5/36 job per hour).
- Time = 1 / (5/36)
- Time = 1 × (36/5) (Dividing by a fraction is the same as multiplying by its reciprocal)
- Time = 36/5 hours
Step 4: Simplify the Answer The fraction 36/5 can be converted to a mixed number or a decimal for clarity.
- 36/5 = 7.2 hours
- In hours and minutes, 0.2 hours is 12 minutes (0.2 × 60 minutes).
- So, the final answer is 7 hours and 12 minutes.
Why This Concept is Crucial for the GRE
The GRE often presents this same logic in more complex, layered questions. * Problems involving "partial work": To give you an idea, "How long would it take to complete half the job together?You might see:
- Problems with three or more workers: The principle remains the same—add all the individual rates.
- Problems where one worker starts later: You would calculate the work done by the first worker alone before the second joins, then use the combined rate for the remaining work. " You would simply use Work = 1/2 in the final step.
It's the bit that actually matters in practice Surprisingly effective..
Common Pitfalls to Avoid
- Averaging the Times: A common mistake is to average 12 and 18 hours to get 15 hours. This is incorrect because it ignores the fact that the faster machine contributes more to the combined effort. The combined time will always be less than the time of the faster machine (i.e., less than 12 hours).
- Adding the Times Instead of the Rates: You cannot add the times directly (12 + 18 = 30) and then divide by 2. The operation must be performed on the rates, not the times.
- Forgetting the Reciprocal: The final step of dividing 1 by the combined rate (5/36) is where many errors occur. Always remember to flip the fraction.
Practice Makes Perfect: Try Another One!
To solidify your understanding, try this variation on your own:
Practice Question: Pipe X can fill a swimming pool in 8 hours. Pipe Y can fill the same pool in 12 hours. If both pipes are opened simultaneously, how long will it take to fill the pool?
Attempt the problem using the steps outlined above. Check your answer by continuing below.
(Solution hidden for practice)
- Rate of X = 1/8 pool per hour
- Rate of Y = 1/12 pool per hour
- Combined Rate = 1/8 + 1/12 = 3/24 + 2/24 = 5/24 pool per hour
- Time = 1 / (5/24) = 24/5 = 4.8 hours (or 4 hours and 48 minutes).
Conclusion
Mastering combined work rate problems is a non-negotiable skill for a high GRE score. By internalizing the Rate = Work / Time formula and consistently applying the principle of adding rates, you transform a potentially confusing word problem into a straightforward algebraic calculation. Remember to practice these steps regularly, watch out for common traps, and approach each question with a clear, methodical plan. With dedicated effort, you’ll find that these problems become some of the most reliable points on your GRE quant section.
Keep practicing with our daily questions, and you'll be well on your way to achieving your target score!
Building on the foundation established, let’s examine a scenario that introduces a time delay. Worth adding: suppose Worker P can finish a report in 10 hours, while Worker Q can complete the same report in 15 hours. Day to day, worker P begins alone and works for 4 hours before Worker Q joins the effort. But to find the total time required, first determine how much of the report is completed in the initial 4 hours. Worker P’s rate is ( \frac{1}{10} ) report per hour, so in 4 hours the portion finished is (4 \times \frac{1}{10}= \frac{4}{10}=0.4) of the report. The remaining work is therefore (1-0.Consider this: 4 = 0. Here's the thing — 6) of the report. Still, once Worker Q joins, the combined rate becomes ( \frac{1}{10}+\frac{1}{15}= \frac{3}{30}+\frac{2}{30}= \frac{5}{30}= \frac{1}{6}) report per hour. Because of that, the time needed to finish the remaining 0. This leads to 6 of the report is (0. 6 \div \frac{1}{6}=0.6 \times 6 = 3.6) hours. So naturally, adding the initial 4 hours gives a total duration of 7. 6 hours.
When the task is only partially completed—say, only half of the job is required—the same rate‑addition method applies; simply replace the total work “1” with the fraction representing the portion to be finished. 4) hours. To give you an idea, if the combined rate of two machines is ( \frac{5}{24} ) job per hour and the objective is to finish half the job, the time needed is ( \frac{1/2}{5/24}= \frac{12}{5}=2.This illustrates that the “Work” component of the formula can be any numeric value, not just the whole job.
GRE‑specific tactics can further streamline these problems. In practice, next, translate each sentence into a rate: “can do the job in X hours” becomes ( \frac{1}{X} ) job per hour. Here's the thing — first, identify the constant amount of work (the whole job) and ignore any irrelevant numbers such as the color of a machine or the temperature in the room. Because of that, then, decide whether the rates should be added (simultaneous work), subtracted (work being undone), or used in a sequence (one worker starts later). Finally, solve for the unknown time and verify that the answer is less than the fastest individual time, as the combined effort must be more efficient.
Our platform supplements these concepts with timed drills that mimic GRE conditions, video walkthroughs that break down each step, and a community forum where you can request personalized feedback on your setups. Regular practice with varied scenarios—three workers, delayed starts, and partial completion—will cement the pattern‑recognition skill set that the test rewards.
By converting every time reference into a rate, adding or adjusting rates as the situation demands, and carefully managing the “Work” component, you turn even the most tangled combined‑work questions into routine calculations. Mastery of these techniques ensures that you allocate time wisely during the exam and convert potential obstacles into reliable points toward your target score Worth keeping that in mind..
