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14 Jul 2025, 12:34
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muditbansal
The equation (r - 2)(t + 4) = 336 is correct because Don was still paid the same total amount, $336. He worked longer and earned less per hour, but the overall pay didn’t change. So we equate it to 336, not to any additional amount.
13 Oct 2025, 09:32
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Understanding What's Really Happening
Don quotes a job for $336 based on his regular hourly rate and estimated time. Here's the twist: the job ends up taking 4 hours longer than he estimated, but he still gets paid the same $336. Because he worked longer for the same total payment, his effective hourly rate dropped by $2.
The question asks: How many hours did Don originally estimate?
Step 1: Set Up Your Variables
Let's define what we're working with:
Step 2: Write Your Constraint Equations
Here's the key insight you need to see: Rate × Time = Total Payment applies to both scenarios, and that total payment is $336 in both cases.
Original estimate: \(r \times t = 336\)
What actually happened: \((r - 2) \times (t + 4) = 336\)
Notice how both expressions equal 336? This is your golden constraint.
Step 3: Solve by Substitution
From the first equation, you can express \(r\) in terms of \(t\):
\(r = \frac{336}{t}\)
Now substitute this into the second equation:
\((\frac{336}{t} - 2) \times (t + 4) = 336\)
Let's expand this carefully:
\(\frac{336(t + 4)}{t} - 2(t + 4) = 336\)
\(336 + \frac{1344}{t} - 2t - 8 = 336\)
Simplifying:
\(\frac{1344}{t} - 2t - 8 = 0\)
Multiply through by \(t\) to clear the fraction:
\(1344 - 2t^2 - 8t = 0\)
Rearrange to standard form:
\(2t^2 + 8t - 1344 = 0\)
Divide everything by 2:
\(t^2 + 4t - 672 = 0\)
Step 4: Factor the Quadratic
You need two numbers that multiply to \(-672\) and add to \(4\).
Think about it: \(28 \times 24 = 672\), and \(28 - 24 = 4\) ✓
So the factored form is: \((t + 28)(t - 24) = 0\)
This gives you \(t = -28\) or \(t = 24\)
Since time cannot be negative, \(t = 24\) hours.
Quick Verification:
If the estimated time was 24 hours for $336 total, his regular rate was \(\frac{336}{24} = 14\) dollars per hour.
The actual job took \(24 + 4 = 28\) hours at a rate of \(14 - 2 = 12\) dollars per hour.
Check: \(28 \times 12 = 336\) ✓
Answer: (B) 24
---
The solution above walks you through the core algebra, but mastering rate-time-work problems requires understanding the systematic framework and recognizing common variations. The complete solution on Neuron shows you the strategic approach that applies to all similar problems, the common trap answers the GMAT uses (and why students fall for them), and time-saving pattern recognition techniques. You can check out the detailed explanation on Neuron by e-GMAT to master these concepts systematically. You can also explore comprehensive solutions for other GMAT official questions on Neuron with practice quizzes and detailed analytics into your strengths and weaknesses.
Don quotes a job for $336 based on his regular hourly rate and estimated time. Here's the twist: the job ends up taking 4 hours longer than he estimated, but he still gets paid the same $336. Because he worked longer for the same total payment, his effective hourly rate dropped by $2.
The question asks: How many hours did Don originally estimate?
Step 1: Set Up Your Variables
Let's define what we're working with:
- \(t\) = estimated time in hours (this is what we need to find)
- \(r\) = regular hourly rate in dollars
- Actual time = \(t + 4\) hours
- Actual rate = \(r - 2\) dollars per hour
Step 2: Write Your Constraint Equations
Here's the key insight you need to see: Rate × Time = Total Payment applies to both scenarios, and that total payment is $336 in both cases.
Original estimate: \(r \times t = 336\)
What actually happened: \((r - 2) \times (t + 4) = 336\)
Notice how both expressions equal 336? This is your golden constraint.
Step 3: Solve by Substitution
From the first equation, you can express \(r\) in terms of \(t\):
\(r = \frac{336}{t}\)
Now substitute this into the second equation:
\((\frac{336}{t} - 2) \times (t + 4) = 336\)
Let's expand this carefully:
\(\frac{336(t + 4)}{t} - 2(t + 4) = 336\)
\(336 + \frac{1344}{t} - 2t - 8 = 336\)
Simplifying:
\(\frac{1344}{t} - 2t - 8 = 0\)
Multiply through by \(t\) to clear the fraction:
\(1344 - 2t^2 - 8t = 0\)
Rearrange to standard form:
\(2t^2 + 8t - 1344 = 0\)
Divide everything by 2:
\(t^2 + 4t - 672 = 0\)
Step 4: Factor the Quadratic
You need two numbers that multiply to \(-672\) and add to \(4\).
Think about it: \(28 \times 24 = 672\), and \(28 - 24 = 4\) ✓
So the factored form is: \((t + 28)(t - 24) = 0\)
This gives you \(t = -28\) or \(t = 24\)
Since time cannot be negative, \(t = 24\) hours.
Quick Verification:
If the estimated time was 24 hours for $336 total, his regular rate was \(\frac{336}{24} = 14\) dollars per hour.
The actual job took \(24 + 4 = 28\) hours at a rate of \(14 - 2 = 12\) dollars per hour.
Check: \(28 \times 12 = 336\) ✓
Answer: (B) 24
---
The solution above walks you through the core algebra, but mastering rate-time-work problems requires understanding the systematic framework and recognizing common variations. The complete solution on Neuron shows you the strategic approach that applies to all similar problems, the common trap answers the GMAT uses (and why students fall for them), and time-saving pattern recognition techniques. You can check out the detailed explanation on Neuron by e-GMAT to master these concepts systematically. You can also explore comprehensive solutions for other GMAT official questions on Neuron with practice quizzes and detailed analytics into your strengths and weaknesses.
13 Oct 2025, 10:39
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So basically the values that we put in should fulfil the equation - T= 2r - 4
Bunuel
