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655-705 (Hard)|   Word Problems|                                 
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mcelroytutoring
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At his regular hourly rate, Don had estimated the labor cos 26 May 2016, 17:33
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This is one of the toughest questions in the OG: 85% incorrect on GMAT Club!

I have found that this one is much easier if you just draw a factor tree and use a little bit of trial and error with the various factors (and combinations thereof). Look for one set of numbers that is 4 apart, and the other 2 apart, and you have your answer.

You know that the rate and the time are going to be relatively close to one another, because a difference of $2 in Don's hourly rate results in a difference of 4 hours in the time spent on the job. These numbers are not exactly the same, but they are close, suggesting that r and t are relatively close to one another in value.

Attached is a visual that should help.
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Screen Shot 2016-05-26 at 5.33.02 PM.png [ 108.82 KiB | Viewed 11436 times ]

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At his regular hourly rate, Don had estimated the labor cos 15 May 2019, 04:06
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Let's talk strategy here. Many explanations of Quantitative questions focus blindly on the math, but remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. As a result, I might write out some steps that I would normally just do in my head on the GMAT, but I want to make sure everyone sees the complete approach. Ready? Here is the full "GMAT Jujitsu" for this question:

First, the problem obviously gives us different rates and times. Since \(Amount = Rate * Time\), the initial situation can be expressed as \(336 = R*T\). The adjusted situation (using \(R\) and \(T\) to represent the initial amounts) would be \(336 = (R-2)(T+4)\). Since this is a system of two equations, we can substitute chunks of one to eliminate what we don’t want to see in the other. There is a natural temptation to substitute the \(R*T\) in the first equation for the \(336\) in the other. However, this doesn’t get rid of either variable – in fact, it gets rid of a number. Substituting in this way would only get us down to a smaller equation, but still with two variables. Our “target” for this equation is \(T\), so let’s eliminate \(R\).

If \(336 = R*T\), then \(R = \frac{336}{T}\) and \(336=(\frac{336}{T}-2)(T+4)\)

Multiplying both sides by \(T\) gives us: \(336T=(336-2T)(T+4)\)

Foiling this out gives us: \(336T=336T+336(4)-2T^2-8T\)

This simplifies to: \(T^2+4T=2(336)\)

At this stage, there is another temptation to bring everything over to one side so we can have it in simplified quadratic form. However, factoring this quadratic might be a little messy. I like to call math like this “Mathugliness” in my classes. Whenever it looks like the math is getting ugly, try looking for other, more conceptual ways to approach the problems. After all, the GMAT is critical-thinking test, not a “let’s-see-if-you-can-solve-obnoxious-math-the-long-way-around” test. Incidentally, notice that I didn’t multiply large numbers out when I simplified the equations above. Multiplying just makes bigger numbers, so I avoid that until I have to. This not only saves time, it also allows me to more easily see factors.

Now, one underutilized tool for many test takers is what I call “Look Out Below!” Use your answer choices as assets you can leverage to think about the question. With this problem, we have really pretty, integer answers that need to be plugged into otherwise messy math. And one of them is the right answer. You can use this to your advantage.

Since our equation above simplifies down to \(T(T+4)=2(336)\), we can plug in values into our equation, and see which ones work.

In this case, it is really easy. Answer choice “C” is way too small, since \((16)(16+4) = 16*20=320\). “D” and “E” resolve to even small numbers. All three answers can be quickly eliminated. Answer choice "A" also fails, but for a different reason: The units digit if \(T=28\) doesn’t even work out in our formula \((T)(T+4) = 2(336)\). (The units digit of 28*32 is "6", but the units digit of 2*336 is "2". You don't need to do any more math than that!)

The only answer remaining is "B", which is the right answer. On the GMAT, I wouldn't even look any further. However, it might be useful here to prove why "B" works, from a critical-thinking perspective. If \(T=24\), then \(24(28)\) should be equal to \(2(336)\). Instead of multiplying this out – which just makes bigger numbers and takes unnecessary time – it is easier to factor the numbers, trying to turn \(2(336)\) into \(24(28)\). Our goal, therefore, is to factor out a \(12\) out of \(336\). This is a piece of cake. After all, \(336=6(50+6)=6(2)(25+3) = 12*28\). And \(24(28) = 2(12*28)\). We are done without doing any messy multiplication. It matches perfectly. “B” is definitely our answer.

Now, let’s look back at this problem from the perspective of strategy. This question can teach us several patterns seen throughout the GMAT. First, the GMAT tries to bait you into doing math the long way around. But if you use the answer choices as part of the analysis of the problem, look for common factors, and intelligently use math in a strategic way, you can avoid a lot of "Mathugliness!" That is how you think like the GMAT.
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Can you explain to solve this problem conceptually? Not even without the math or equations. I've been a fan of Veritas prep books where solutions to problems are only conceptual sometimes, not even with equations. That's so amazing. If there is any conceptual related solution, please add it here. Thanks in advance!
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At his regular hourly rate, Don had estimated the labor cos 18 May 2019, 14:48
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Saiganesh999

Can you explain to solve this problem conceptually? Not even without the math or equations. I've been a fan of Veritas prep books where solutions to problems are only conceptual sometimes, not even with equations. That's so amazing. If there is any conceptual related solution, please add it here. Thanks in advance!
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Great question, Saiganesh999! I am so glad that the Veritas Prep solutions resonate with you. Most people end up spinning their wheels trying to do the math the long way around because they don't realize how conceptually-oriented many Quantitative GMAT questions really are. Conceptually looking at a question can sometimes turn a mean, 5-minute problem into a quick, 20-second solution. However, it is important to note that not every question can be solved solely by waving a conceptual wand and having the answer fall into your lap. I would even argue that most GMAT Quantitative questions are a combination of both conceptual and mathematical approaches. (And many conceptual approaches only become visible once you get a couple of steps into the math!)

This problem is no exception. If you look at my solution (https://gmatclub.com/forum/at-his-regul ... l#p2143221), you can see that once we distill the equation down to one variable, then we can look at it conceptually -- or more specifically, we can use our answer choices as leverage to think about the question. (I call this strategy "Look Out Below!" in my classes.) We can quickly eliminate answer choices C, D, and E because they are too small. Answer choice A can be eliminated because it fails the unit's digit test. Approximation, unit's digit math, and plugging in answer choices are all part of a comprehensive, balanced strategy you will want to use on the GMAT.
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At his regular hourly rate, Don had estimated the labor cos 11 Sep 2020, 08:16
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At his regular hourly rate, Don had estimated the labour cost of a repair job as $336 and he was paid that amount. However, the job took 4 hours longer than he had estimated and, consequently, he earned $2 per hour less than his regular hourly rate. What was the time Don had estimated for the job, in hours?

(A) 28
(B) 24
(C) 16
(D) 14
(E) 12

Say the regular hourly rate was \(r\)$ and estimated time was \(t\) hours, then we would have:

\(rt=336\) and \((r-2)(t+4)=336\);

So, \((r-2)(t+4)=rt\) --> \(rt+4r-2t-8=rt\) --> \(t=2r-4\).

Now, plug answer choices for \(t\) and get \(r\). The pair which will give the product of 336 will be the correct answer.

Answer B fits: if \(t=24\) then \(r=14\) --> \(rt=14*24=336\).

Answer: B.

Hope it's clear.
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My confusion is: Will not the estimated repair cost be changed?
I solved the question with the tougher way:
(336/T)- (336/T+4) = 2
It needs lots of calculations.
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At his regular hourly rate, Don had estimated the labor cos 11 Sep 2020, 19:46
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Hi Baten80,

The first sentence of the prompt tell us that Don estimated the cost as $336 and was PAID that amount. The additional information tells us that Don had to work MORE hours and was thus paid LESS per HOUR overall - meaning that the cost did NOT change.

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At his regular hourly rate, Don had estimated the labor cos Updated on: 31 May 2021, 07:14
Originally posted by mcelroytutoring on 30 May 2021, 14:55.
Last edited by mcelroytutoring on 31 May 2021, 07:14, edited 4 times in total.
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The quadratic algebra can get a little complicated here, and the factor tree method I suggested 5 years ago is not always ideal, so I would also suggest straightforward backsolving (trying the answers and adjusting accordingly) as an alternative, easier way to solve this problem.

What is backsolving? Essentially, it involves trying your answers to see whether the rest of the numerical details in the question stem fall into place, then adjusting in the correct direction (higher or lower) when necessary. If you adjust in the correct direction, then you should never have to try more than 2 answers.

Let's start in the middle with Choice C and test. If Don had estimated 16 hours, then he would be making $336/16 = $21/hour. However, the question tells us that he actually worked 4 hours more than the estimate, at $2/less, which in this case would have been 20 hours x $19/hour = $380. That's too much (the actual total was $336), so he must have estimated more hours at a lower rate. Try a higher number for the estimate, then: 24 hours.

If he estimated 24 hours, then he would be making $336/24 = $14/hour. The question tells us that he actually worked 4 hours more than the estimate, at $2/less, which in this case would have been 28 hours x $12/hr = $336.

This is the correct total, so Choice B is correct.

-Brian
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At his regular hourly rate, Don had estimated the labor cos 11 Jun 2021, 06:56
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Video solution from Quant Reasoning starts at 0:26:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
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At his regular hourly rate, Don had estimated the labor cos 27 Mar 2022, 07:32
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Video solution from Quant Reasoning starts at 0:26:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
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avigutman

This is so helpful, thank you!!!
Initially, I got stuck using this method because I said wait the prime factorization of 28 has an extra 2, so when 24 and 28 are combined, the prime factors of those two numbers added does not exactly equal the prime factors of 336... so that cannot be my answer. Why aren't we looking for an exact match here in prime factors?
Thank you again.
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At his regular hourly rate, Don had estimated the labor cos 27 Mar 2022, 12:39
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woohoo921

Initially, I got stuck using this method because I said wait the prime factorization of 28 has an extra 2, so when 24 and 28 are combined, the prime factors of those two numbers added does not exactly equal the prime factors of 336... so that cannot be my answer. Why aren't we looking for an exact match here in prime factors?
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woohoo921 if I understood you correctly, you multiplied 24*28 (24 is 2^3 * 3^1, and 28 is 2^2 * 7^1, so their product is 2^5 * 3^1 * 7^1).
Is that what you did? If so, that was your mistake. 24 and 28 independently need to be factors of 336, but not simultaneously.
As an analogy, 6 is a factor of 12 and 4 is a factor of 12, but 6*4=24 is not a factor of 12.
Multiplying 24 by 28 in this problem isn't necessary. Rather, this problem plays on the equivalence of 24*14 and 28*12.
Does that help?
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At his regular hourly rate, Don had estimated the labor cos 01 Apr 2022, 19:48
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woohoo921

Initially, I got stuck using this method because I said wait the prime factorization of 28 has an extra 2, so when 24 and 28 are combined, the prime factors of those two numbers added does not exactly equal the prime factors of 336... so that cannot be my answer. Why aren't we looking for an exact match here in prime factors?

woohoo921 if I understood you correctly, you multiplied 24*28 (24 is 2^3 * 3^1, and 28 is 2^2 * 7^1, so their product is 2^5 * 3^1 * 7^1).
Is that what you did? If so, that was your mistake. 24 and 28 independently need to be factors of 336, but not simultaneously.
As an analogy, 6 is a factor of 12 and 4 is a factor of 12, but 6*4=24 is not a factor of 12.
Multiplying 24 by 28 in this problem isn't necessary. Rather, this problem plays on the equivalence of 24*14 and 28*12.
Does that help?
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To clarify, from my understanding your method was that the prime factorization of 336 is= (2^4)(3^1)(7^1). Then, you found the prime factorization of 24 which is (2^3)(3^1). You then said that you are missing (2^2)(2^7) and that the missing primes come from the prime box of 28. The prime box of 28 is then (2^2)(7). Correct? So basically it all evens out?
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At his regular hourly rate, Don had estimated the labor cos 02 Apr 2022, 04:53
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woohoo921
Then, you found the prime factorization of 24 which is (2^3)(3^1). You then said that you are missing (2^2)(2^7) and that the missing primes come from the prime box of 28. The prime box of 28 is then (2^2)(7). Correct? So basically it all evens out?
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No, woohoo921, at 8:30 I say that once we've accounted for 24, we still need a (2^1)(7^1) to get to 336. That's 14, not 28.
Then, separately I find the prime box of 28 (2^2)(7^1) and discover that I'd need a (2^2)(3^1) to get to 336. That's 12.
In conclusion, answer choice B has us getting to 336 with 24*14 vs with 28*12, and these two products are a match for the description in the question stem (28 is 4 more than 24, and 12 is 2 less than 14).
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At his regular hourly rate, Don had estimated the labor cos 27 Jun 2022, 09:39
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At his regular hourly rate, Don had estimated the labor cost of a repair job as $336 and he was paid that amount. However, the job took 4 hours longer than he had estimated and, consequently, he earned $2 per hour less than his regular hourly rate. What was the time Don had estimated for the job, in hours?

(A) 28
(B) 24
(C) 16
(D) 14
(E) 12
Show more

We can Plug In The Answers (PITA). I like trying B and D.

B: If it had taken the time expected, he'd have made 336/24 = $14 per hour. It took 28 hours, so he made 336/28 = $12 per hour. Is that what we wanted? Yep!

Answer choice B.


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At his regular hourly rate, Don had estimated the labor cos 25 Aug 2022, 06:03
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Can someone please explain the question? How did he receive the same amount when the hours are increased?
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At his regular hourly rate, Don had estimated the labor cos 25 Aug 2022, 16:34
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Can someone please explain the question? How did he receive the same amount when the hours are increased?
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Hi aksh123456123456,

The first sentence of the prompt tell us that Don estimated the cost as $336 and was PAID that amount. However, the additional information tells us that Don had to work MORE hours than he originally estimated the job would take - but since he was paid the estimated cost ($336), he was thus paid LESS per HOUR overall.

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At his regular hourly rate, Don had estimated the labor cos 07 Sep 2022, 11:19
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Is it just me or does this not feel like a 2min question ? Let’s assume you made it to the quadratic and then started substitution of answers , still takes a lot of time .

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At his regular hourly rate, Don had estimated the labor cos 07 Sep 2022, 18:07
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Is it just me or does this not feel like a 2min question ? Let’s assume you made it to the quadratic and then started substitution of answers , still takes a lot of time .

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Hi SignUp,

Most GMAT questions are written so that they can be approached in more than one way - and in this prompt, working through a math-heavy approach using Quadratics would likely take you far more than 2 minutes (you can actually see from the 'stats' for this question that lots of people were not able to solve it in 2 minutes - or even close to it). There's actually a much faster way to answer this question that does not require a Quadratic at all - but you still have to do some basic multiplication and division to find the correct answer. As an aside, the 'goal' to answer every Quant question in 2 minutes or less is a terrible idea - and if you are performing at a high level, then you will absolutely see some Quant questions on the Official GMAT that might take you upwards of 3 minutes to solve.

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At his regular hourly rate, Don had estimated the labor cos 23 Dec 2022, 12:44
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$/hr = $336/hours worked.

Regular hours worked- 336/x

current hours worked - (regular hours worked + 4) -----336/x + 4

$x-2 = 336 / (336/x + 4)

FOIL:
336 + 4X- 672/X - 8 = 336

subtract 336 from both sides and multiply by x

4x^2- 672 - 8x = 0

divide through by 4

x^2 - 2x - 168 = 0

Look through the options for two numbers that add to that multiply to -168 and see if you can manipulate them to also add to -2.

Your two numbers are 14 and 12.

(x-14) (x+12) = 0.

x cant be negative so x =14.

Now the question is for the number of hours. remember that x is dollar per hour.

336/14 = 24 hours worked.
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At his regular hourly rate, Don had estimated the labor cos 28 Mar 2025, 05:59
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But e) T=12 also satisfies as per your approach. T= 12, T+4 = 16 , both are factors of 336, and R = 23
KarishmaB
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At his regular hourly rate, Don had estimated the labour cost of a repair job as $336 and he was paid that amount. However, the job took 4 hours longer than he had estimated and, consequently, he earned $2 per hour less than his regular hourly rate. What was the time Don had estimated for the job, in hours?

(A) 28
(B) 24
(C) 16
(D) 14
(E) 12

You utilize the answer choices whenever you CAN. Here I would keep an eye on the choices right from the start. I would say
R*T = 336 (his regular hourly rate * time he estimated)
The options give us the value of T which is an integer.

\(336 = 2^4*3*7\)

So R*T = 336
(R-2)*(T + 4) = 336
So T as well as T+4 should be factors of 336.
If T is 28, T+4 is 32 which is not a factor of 336 so ignore it.
If T is 24, T+4 is 28. Both are factors of 336. Keep it. If T is 24, R is 14. So (R - 2) is 12. 12*28 does gives us 336 so T = 24 must be the correct answer.

But note that if you want to reduce your mechanical work, you need to be fast in your calculations. You cannot spend a minute working on every option or making calculation mistakes.
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DanTheGMATMan
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At his regular hourly rate, Don had estimated the labor cos 18 Jun 2025, 21:39
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There is a better way folks:

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muditbansal
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At his regular hourly rate, Don had estimated the labor cos 14 Jul 2025, 01:53
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Hi Bunuel,

I just have one doubt why did we equate (r-2) * (t+4) = 336. If we are multiplying the new rate and longer time the result would be other than 336 right? It will be the additional amount that Don earned. Please let me know if I am thinking correctly
Bunuel
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At his regular hourly rate, Don had estimated the labour cost of a repair job as $336 and he was paid that amount. However, the job took 4 hours longer than he had estimated and, consequently, he earned $2 per hour less than his regular hourly rate. What was the time Don had estimated for the job, in hours?

(A) 28
(B) 24
(C) 16
(D) 14
(E) 12

Say the regular hourly rate was \(r\)$ and estimated time was \(t\) hours, then we would have:

(i) \(rt=336\).

(ii) \((r-2)(t+4)=336\) (we are told that the job took 4 hours longer than he had estimated and, consequently, he earned $2 per hour less than his regular hourly rate, thus \((r-2)(t+4)=336\));

So, \((r-2)(t+4)=rt\);

\(rt+4r-2t-8=rt\);

\(t=2r-4\).

Now, plug answer choices for \(t\) and get \(r\). Finally, since we know that \(rt=336\), then the pair which will give the product of 336 will be the correct answer.

Answer B fits: if \(t=24\) then \(r=14\) and \(rt=14*24=336\).

Answer: B.

Hope it's clear.
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