OG problem: a fast way to factor?

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massi2884: Senior | Next Rank: 100 Posts; Posts: 95; Joined: Tue Sep 13, 2011 3:19 pm; Thanked: 3 times; GMAT Score:710

by massi2884 » Tue Apr 10, 2012 9:22 am

[spoiler]OA is B However could you please explain a fast way to go from t^2 +4t -672 = 0 to (t-24)(t+28) ? I spent a lot of time finding the two values 28 and 24, is there a fast way to factor?[/spoiler]

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?

28
24
16
14
12

[spoiler]OA is B However could you please explain a fast way to go from t^2 +4t -672 = 0 to (t-24)(t+28) ? I spent a lot of time finding the two values 28 and 24, is there a fast way to factor?[/spoiler]

Thanks.

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Source: Beat The GMAT — Problem Solving | Tue Apr 10, 2012 9:22 am

Bill@VeritasPrep: GMAT Instructor; Posts: 1248; Joined: Thu Mar 29, 2012 2:57 pmFollowed by: 192 members; Location: Everywhere; Thanked: 503 times; GMAT Score:780

by Bill@VeritasPrep » Tue Apr 10, 2012 10:11 am

I think the fastest way to solve is to avoid the quadratic. We can set up two equations (r=hourly rate, h=hours)

Theoretical job: 336=rh
Actual job: 336=(r-2)(h+4)

We can expand that second equation:

336 = rh + 4r - 2h -8

Conveniently, we now have an rh term, which the first equation tells us is equal to 336. Let's substitute:

336 = 336 + 4r - 2h - 8
0 = 4r - 2h - 8
8 = 4r -2h
4 = 2r - h

h=2r - 4

At this point, I started working with the answer choices, which give us values for h. If h=16, then the rate is $10. Does $10 * 16 = 336? No, so our h value must be larger, leaving A and B as possibilities.

If h=24, then the rate is $14. Does $24 * 14 = 336? Yes. That means that B is the correct answer.

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shubham_k: Junior | Next Rank: 30 Posts; Posts: 27; Joined: Sun Apr 08, 2012 5:38 am; Thanked: 20 times

by shubham_k » Tue Apr 10, 2012 10:17 am

the immediate factors of 336 can be formed and they are, (2^4)*3*7. Thus we see that 28 16 and 14 cannot be the answer as 32 (28+4) (2^5), 20 (16+4) (2*2*5)and 18 (14+4) (3*3*2) are not factors of 336.Now only 24 and 12 are remaining and we can now without much effort can find the answer to be 24.

Please note this an alternate method apart from the normal unitary method approach by assuming the hourly wage to be x and number of hours to be t and forming basic equations and solving.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Apr 10, 2012 10:21 am

massi2884 wrote:[spoiler]OA is B However could you please explain a fast way to go from t^2 +4t -672 = 0 to (t-24)(t+28) ? I spent a lot of time finding the two values 28 and 24, is there a fast way to factor?[/spoiler]

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?

28
24
16
14
12

[spoiler]OA is B However could you please explain a fast way to go from t^2 +4t -672 = 0 to (t-24)(t+28) ? I spent a lot of time finding the two values 28 and 24, is there a fast way to factor?[/spoiler]

Thanks.

We can plug in the answers, which represent the estimated time: 28, 24, 16, 14, or 12.
The actual time was 4 hours longer: 32, 28, 20, 18, or 16.
Since 336 is not a multiple of 32, eliminate A.
Since 336 is not a multiple of 20, eliminate C.
336 is not a multiple of 9, since the sum of its digits is not a multiple of 9.
Thus, 336 is not a multiple of 18; eliminate D.

The correct answer must be B or E.

Answer choice B: estimated time = 24, actual time = 28
Estimated payrate = 336/24 = 14.
Actual payrate = 336/28 = 12.
Difference = 14-12 = 2.
Success!

The correct answer is B.

Note that we had to try only ONE answer choice -- a very efficient way to solve the problem.

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Birottam Dutta: Master | Next Rank: 500 Posts; Posts: 342; Joined: Wed Jul 08, 2009 8:50 amFollowed by: 19 members; Thanked: 214 times; GMAT Score:740

by Birottam Dutta » Wed Apr 11, 2012 6:23 am

The fastest way of solving such problems is to straightaway check the options and see which of them fits the given condition.

Divide 336 by the options 28, 24, 16, 14 and 12.

We get, 12, 14, 21, 24 and 28. These are the values for the hourly wages. In these wages, we see that only 12 and 14 have a difference of 2, which is the requirement of the question statement.

Hence, as per the problem statement, the initial estimated hours were 24 at a wage of 14 per hour. It took 28 hours and so hourly wage became 12.

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