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At his regular hourly rate, Don had estimated the labor cos

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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 26 Nov 2017, 12:25
Can anyone please comment on the following strategy, given that I was able to write down the proper equations (i.e. rt= 336 and (r-2)(t+4) = 336).

I almost immediately realized that I wouldn't be able to solve that in under 4-5 minutes with my current ability in algebra, so I went for pairing...

Considering one of the answers will give me the time "t", and considering the GMAT loves traps, I looked for a combination of r*t that would give me 336, which wasn't too hard looking at the unit digits...


I fully understand I should practice on getting this solved in about 2 minutes, but this is not where I'm at currently. So on test day I would be left with the option of tanking the test by spending 5-6 minutes or randomly guessing. Please let me know your thoughts.

Thanks!
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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 26 Dec 2017, 05:39
Bunuel wrote:
macjas wrote:
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.


Hello Bunuel, here is my solution

let total number of hours be y
let hourly rate be x
Total amount payed x*y= 360 ---> Hourly rate x= y/336
to complete job it took 4 hours longer --> y+4
hourly rate reduced by 2 USD ---> x-2

(y+4)(x-2) = 336
plug in into above equation x= y/336

(y+4)(y/336 - 2)

336y/y - 2y+1344/y - 8 = 336
336 -2y+1344/y-8=336
-2y+1344/y= 336+8-336
-2y+1344/y =8
ok after this I got stuck and confused, where am I going what I am solving :)

can you please explain what have I done wrong ? I decoded the info correctly I mean expressed initially in numbers.

thanks for your help.
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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 26 Dec 2017, 06:57
dave13 wrote:
Bunuel wrote:
macjas wrote:
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.


Hello Bunuel, here is my solution

let total number of hours be y
let hourly rate be x
Total amount payed x*y= 360 ---> Hourly rate x= y/336
to complete job it took 4 hours longer --> y+4
hourly rate reduced by 2 USD ---> x-2

(y+4)(x-2) = 336
plug in into above equation x= y/336

(y+4)(y/336 - 2)

336y/y - 2y+1344/y - 8 = 336
336 -2y+1344/y-8=336
-2y+1344/y= 336+8-336
-2y+1344/y =8
ok after this I got stuck and confused, where am I going what I am solving :)

can you please explain what have I done wrong ? I decoded the info correctly I mean expressed initially in numbers.

thanks for your help.


If \(xy = 336\), then \(x = \frac{336}{y}\), not x = y/336.

\((y+4)(\frac{336}{y} - 2) = 336\);

\(336 - 2y + 4*\frac{336}{y} - 8 = 336\);

\(y^2 + 4y - 672 = 0\);

\(y(y + 4) = 672\)

Plug options: \(y = 24\) works.

Hope it helps.
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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 26 Dec 2017, 08:50
here is my solution

let total number of hours be y
let hourly rate be x
Total amount payed x*y= 360 ---> Hourly rate x= y/336
to complete job it took 4 hours longer --> y+4
hourly rate reduced by 2 USD ---> x-2

(y+4)(x-2) = 336
plug in into above equation x= y/336

(y+4)(y/336 - 2)

336y/y - 2y+1344/y - 8 = 336
336 -2y+1344/y-8=336
-2y+1344/y= 336+8-336
-2y+1344/y =8
ok after this I got stuck and confused, where am I going what I am solving :)

can you please explain what have I done wrong ? I decoded the info correctly I mean expressed initially in numbers.

thanks for your help.[/quote]

If \(xy = 336\), then \(x = \frac{336}{y}\), not x = y/336.

\((y+4)(\frac{336}{y} - 2) = 336\);

\(336 - 2y + 4*\frac{336}{y} - 8 = 336\);

\(y^2 + 4y - 672 = 0\);

\(y(y + 4) = 672\)

Plug options: \(y = 24\) works.

Hope it helps.[/quote]

Bunuel - Many thanks ! Just two more question

could you show in detail how you got this \(y^2 + 4y - 672 = 0\); and another question what is the point of factoring ? \(y(y + 4) = 672\) :? I remember when we see such equation\(y^2 + 4y - 672 = 0\); we need to find discriminant ? like D = B^2-4AC
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New post 26 Dec 2017, 08:58
dave13 wrote:
Bunuel - Many thanks ! Just two more question

could you show in detail how you got this \(y^2 + 4y - 672 = 0\); and another question what is the point of factoring ? \(y(y + 4) = 672\) :? I remember when we see such equation\(y^2 + 4y - 672 = 0\); we need to find discriminant ? like D = B^2-4AC


\(336 - 2y + 4*\frac{336}{y} - 8 = 336\);

Cancel 336: \(- 2y + 4*\frac{336}{y} - 8 = 0\);

Reduce by 2: \(- y + 2*\frac{336}{y} - 4 = 0\);

Multiply by y: \(-y^2+672-4y=0\)

Re-arrange: \(y^2 + 4y - 672 = 0\);

Here you can solve for y with discriminant formula but it's easier to factor the way shown and PLUG OPTIONS.
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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 26 Dec 2017, 09:26
Bunuel wrote:
dave13 wrote:
Bunuel - Many thanks ! Just two more question

could you show in detail how you got this \(y^2 + 4y - 672 = 0\); and another question what is the point of factoring ? \(y(y + 4) = 672\) :? I remember when we see such equation\(y^2 + 4y - 672 = 0\); we need to find discriminant ? like D = B^2-4AC


\(336 - 2y + 4*\frac{336}{y} - 8 = 336\);

Cancel 336: \(- 2y + 4*\frac{336}{y} - 8 = 0\);

Reduce by 2: \(- y + 2*\frac{336}{y} - 4 = 0\);

Multiply by y: \(-y^2+672-4y=0\)

Re-arrange: \(y^2 + 4y - 672 = 0\);

Here you can solve for y with discriminant formula but it's easier to factor the way shown and PLUG OPTIONS.


Thank you Bunuel. When you reduce by 2, why didn't you reduce fraction by 2 as well 336/y Normally when we reduce and or multiply numbers in equation all numbers are subject to be changed accordingly - No ? :? please correct me if I am wrong.
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New post 26 Dec 2017, 09:30
dave13 wrote:
Bunuel wrote:
dave13 wrote:
Bunuel - Many thanks ! Just two more question

could you show in detail how you got this \(y^2 + 4y - 672 = 0\); and another question what is the point of factoring ? \(y(y + 4) = 672\) :? I remember when we see such equation\(y^2 + 4y - 672 = 0\); we need to find discriminant ? like D = B^2-4AC


\(336 - 2y + 4*\frac{336}{y} - 8 = 336\);

Cancel 336: \(- 2y + 4*\frac{336}{y} - 8 = 0\);

Reduce by 2: \(- y + 2*\frac{336}{y} - 4 = 0\);

Multiply by y: \(-y^2+672-4y=0\)

Re-arrange: \(y^2 + 4y - 672 = 0\);

Here you can solve for y with discriminant formula but it's easier to factor the way shown and PLUG OPTIONS.


Thank you Bunuel. When you reduce by 2, why didn't you reduce fraction by 2 as well 336/y Normally when we reduce and or multiply numbers in equation all numbers are subject to be changed accordingly - No ? :? please correct me if I am wrong.


No. When you divide \(4*\frac{336}{y}\) you get \(\frac{1}{2}*4*\frac{336}{y}=2*\frac{336}{y}\).
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New post 26 Dec 2017, 09:50
Quote:
Quote:
Thank you Bunuel. When you reduce by 2, why didn't you reduce fraction by 2 as well 336/y Normally when we reduce and or multiply numbers in equation all numbers are subject to be changed accordingly - No ? :? please correct me if I am wrong.


No. When you divide \(4*\frac{336}{y}\) you get \(\frac{1}{2}*4*\frac{336}{y}=2*\frac{336}{y}\).


Bunuel - But isn't 336/y a separate number ? Shouldn't we have done so 1/2 *4 and 1/2 * 336/y ? :?
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New post 26 Dec 2017, 09:55
Quote:
Quote:
dave13 wrote:
Thank you Bunuel. When you reduce by 2, why didn't you reduce fraction by 2 as well 336/y Normally when we reduce and or multiply numbers in equation all numbers are subject to be changed accordingly - No ? :? please correct me if I am wrong.


No. When you divide \(4*\frac{336}{y}\) you get \(\frac{1}{2}*4*\frac{336}{y}=2*\frac{336}{y}\).


Bunuel - But isn't 336/y a separate number ? Shouldn't we have done so 1/2 *4 and 1/2 * 336/y ? :?


No. Let me ask you: what is the value of 4*6 when reduced by 2? Is it 2*6 = 12 or 2*3 = 6? I think you'll benefit if you brush up fundamentals before practicing questions.
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New post 26 Dec 2017, 10:06
Quote:
Quote:
Bunuel wrote:

No. When you divide \(4*\frac{336}{y}\) you get \(\frac{1}{2}*4*\frac{336}{y}=2*\frac{336}{y}\).


Bunuel - But isn't 336/y a separate number ? Shouldn't we have done so 1/2 *4 and 1/2 * 336/y ? :?


No. Let me ask you: what is the value of 4*6 when reduced by 2? Is it 2*6 = 12 or 2*3 = 6? I think you'll benefit if you brush up fundamentals before practising questions.


@Bunuel- thanks for asking me this question :) I think if we reduce 4*6 by 2 the answer will be 12 because 4*6 is one number. I have brushed up fundamentals but sometimes in some PS questions i encounter some technical details i have a vague idea about or simply cant use theory in practice. Thanks a lot for explanation :)
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New post 26 Dec 2017, 10:14
dave13 wrote:
@Bunuel- thanks for asking me this question :) I think if we reduce 4*6 by 2 the answer will be 12 because 4*6 is one number. I have brushed up fundamentals but sometimes in some PS questions i encounter some technical details i have a vague idea about or simply cant use theory in practice. Thanks a lot for explanation :)


Generally \(\frac{1}{x}*(a + b) = \frac{a}{x} + \frac{b}{x}\) but \(\frac{1}{x}*(a*b) = \frac{a*b}{x}\)
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New post 26 Dec 2017, 11:59
Bunuel wrote:
dave13 wrote:
Bunuel - Many thanks ! Just two more question

could you show in detail how you got this \(y^2 + 4y - 672 = 0\); and another question what is the point of factoring ? \(y(y + 4) = 672\) :? I remember when we see such equation\(y^2 + 4y - 672 = 0\); we need to find discriminant ? like D = B^2-4AC


\(336 - 2y + 4*\frac{336}{y} - 8 = 336\);

Cancel 336: \(- 2y + 4*\frac{336}{y} - 8 = 0\);

Reduce by 2: \(- y + 2*\frac{336}{y} - 4 = 0\);

Multiply by y: \(-y^2+672-4y=0\)

Re-arrange: \(y^2 + 4y - 672 = 0\);

Here you can solve for y with discriminant formula but it's easier to factor the way shown and PLUG OPTIONS.


Bunuel could you please tell me in which case should I use factoring and in which case conventional method of finding discriminant ? Thanks a lot answering my questions. Highly appreciated!
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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 21 Jul 2018, 03:42
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Bunuel wrote:
macjas wrote:
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.



Solving Quadric equation is taking too much time. So What we take back solving approach.
RT = 336
and, (R-2)(T+4)=336-----(ii)
If we start back solving from C then we get Time 16 and Rate 21, these values don't satisfy equation (ii)
the given answer choices are increasing so we don't have to try D and E options.
Trying B, Time is 24 and Rate is (334/24)= 14
Thus, Puting the values in equation ii, (14-2)(24+4) = 12 x 28 = 336
Ans B.
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New post 26 Sep 2018, 06:00
Well what I did was basically to prime factorize \(336 = \frac{2^4 * 3 * 7}{x}\)

I then tried \(x = 24 = 2^3 * 3\) back in the equation and ended up with \(2 * 7 = 14\)

Now if we add \(x+4 = 24 + 4 = 28\)
\(28 = 2^2 * 7 = 28\) and that would leave \(2^2 * 3 = 12\)
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New post 15 May 2019, 04:06
AaronPond wrote:
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.


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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New post 18 May 2019, 14:48
Saiganesh999 wrote:
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!


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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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 22 May 2019, 22:13
AaronPond wrote:
Saiganesh999 wrote:
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!


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.


Aaron! That's an amazing explanation!

However, when I looked at the problem little further, I understood that when 4 is added to the original time, it must give a clean fraction, since we are having an integer value of 2 in the question stem. So I started with C, which is 16. 16 gives clean integer when it divides 336 but 16+4=20 doesn't give. I moved up with 24, and checked for 24+4=28. 28 gives a clean integer when it divides 336 and I re-checked for 24 too. Both answers give clean integer values and have a difference of 2 $/hour with their quotients, when they divide 336. It's a simple approach that takes <1 minute. Did I miss anything?
Thanks in advance! Regards
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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 17 Aug 2019, 10:53
macjas wrote:
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



336/x =t (1)

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

substituting t = 336/x in equation 2, we get :

336/x-2 = 336/x +4
solving for x we get two values -12, 14 ignore negative value substitute x=14 in equation (1) to get t = 24
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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 22 Aug 2019, 04:33
macjas wrote:
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


Given:
1. At his regular hourly rate, Don had estimated the labor cost of a repair job as $336 and he was paid that amount.
2. However, the job took 4 hours longer than he had estimated
3. He earned $2 per hour less than his regular hourly rate.

Asked: What was the time Don had estimated for the job, in hours?

Let the time Don had estimated for the job, in hours be x hours and let his hourly rate be h
hx = $336 => h = $336/x
336/(x+4) = h -2
336 = (x+4)(h-2) = (x+4)(336/x -2) = 336 + 4*336/x - 2x - 8
1344/x = 2x + 8
\(x^2 + 4x - 672 = 0\)
\((x+2)^2 = 676 = (26)^2\)
x = 24 hours

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Re: At his regular hourly rate, Don had estimated the labor cos  [#permalink]

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New post 22 Aug 2019, 09:47
let say, x dollars per her rate

so 336/x-2 - 336/4= 4

so x = 14


so total hours = 336/14= 24
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Re: At his regular hourly rate, Don had estimated the labor cos   [#permalink] 22 Aug 2019, 09:47

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