PS 137 - At his regular hourly rate, Don had estimated...

[clarkh85]

Here's an Algebraic Translations question that I'm really struggling with. It involves rates and has a quadratic. I've tried it twice and guessed both times because I can't even get reasonably close to eliminating an answer within 2.5 minutes.

Backtesting is a little confusing, but I suspect it must be best when there are quadratics. Shall I draw some sort of grid to help with backtesting?

I've watched the Sections recording where this problem is worked, but I'm not satisfied. Even during the Sections recording, the instructor said a few alarming things, verbatim:

"this is a tedious question no matter what"
"you're just going to have to be patient with it"
"just grind grind grind"

Using RT = W, you wind up with some quadratic:

T^2 + 4T - 672 = 0

No fun to factor 672. And it's not particularly quick to arrive at the equation either.

And there's no explanation in OG Archer.

All questions should be solvable in 2 minutes, right? This is not even a high numbered problem, indicating that it should not be so difficult. So, what's the quickest way to solve this problem?

The question is:

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

Thank you.

[divineacclivity]

Let me try n explain it to you :)
Let's assume the estimated time was 't' hrs
Estimated rate = $ (336/t) per hr

Actual time taken = t+4 hrs
Actual rate = $ (336/t - 2) per hr

total payment made = actual time taken * actual rate = 336
i.e. (t+4) * (336/6 - 2) = 336
i.e. t^2 + 4t - 672 = 0
i.e. t = -28 or t = 24
discard -ve time i.e. -28; keep t = 24

******* how to factor 672 ******
672 is even, so, keep diving 672 by 2 for as long as you can
i.e. 672/2 = 336; 336/2 = 168 and so on:

2|672
2|336
2|168
2|84
2|42
3|21 [21 immediately strikes 7 * 3, so, the following]
7|7
1|1

Now, quickly make pairs 7 * 3 = 21 and 2 * 2 * 2 * 2 *2 = 32
32 - 21 = 11 --> discard because we need to reach 4

another combination: 7 * 2 * 2 = 28 and 2 * 2 * 2 * 3 = 24
28 - 24 = 4 yippie

t^2 + 4t - 672 = 0
t^2 + 28t - 24t - 672 = 0
t(t+28) -24(t+28) = 0
(t+28) (t-24)=0
=> t = - 28 or t = 24
I hope this helps

[jlucero]

Algebraically, this solution looks right. And if you did want to find a fast way to break down T^2 + 4T - 672 = 0, recognize that you're finding two numbers that are close to one another (4 off) and multiply to get 672. I'd start at 25 (25^2 = 625), and find even numbers around 25 that would go into 672.

That said, I'd also probably do this question by backsolving:

Don estimated:

(A) 28 ($12/hr)
(B) 24 ($14/hr)
(C) 16 ($21/hr)
(D) 14 ($24/hr)
(E) 12 ($28/hr)

If I filled out this entire chart, I'd notice that 28*12 = 12*28, so A&E are opposites, as are B&D. But I'd recognize a pattern that the more hours Don estimated he'd work, the lower his wage would be. So if I needed to find a difference of $2/hr, I could know where to estimate. Also, with the GMAT, the trap answer will often be what Don actually worked, so if a trap answer is likely to be an answer choice, I'll be able to see that by figuring out the rates for each estimated amount of time. And that's exactly what we have. The difference b/w A&B is 4 hours and $2/hr, so I can see that the estimation should be the smaller answer choice, (B).