Two questions from OG 12 Edition

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Two questions from OG 12 Edition

by gmat_mba » Sun Apr 05, 2009 8:43 pm
I was wondering if there is a typo in Problem Solving Answer Explanations for questions 148 and 230. Please remember these questions are from OG 12 Edition.

In Q 148: How did they get "y"in (10(x+"y") + 10y) / (x +y)? Please note this is the third step in the answer.
In Q 230: The answer explanation said 2^17 instead of 2^-17. Is this right?

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by Jose Ferreira » Mon Apr 06, 2009 5:31 pm
Hi,

Neither of these is a typo, though both explanations could certainly contain more discussion of the approach taken.

148.
In this case, they go from 10x + 20y in the numerator to 10(x + y) + 10y in the numerator. If we multiply out the second expression, we will see that 10(x + y) + 10y = 10x + 10y + 10y = 10x + 20y, so it is in fact equal to the first expression.

In doing so, they skip a step in the explanation and do not explain their motivation. It would have been more clear if they had written 10x + 10y + 10y first, then factored to 10(x + y) + 10y.

They are doing this because the denominator contains (x + y), and they are seeking an opportunity to cancel out an (x + y) term from the numerator and denominator.

Another approach to this question is to plug in values for k based on the answer choices:

It is best to start out by multiplying everything through by (x + y). Most students are less likely to make mistakes when fractions are eliminated from a question; for this reason, it is helpful to eliminate denominators.

Once we do this, we are left with 10x + 20y = k(x + y), or 10x + 20y = kx + ky.

One thing you might notice about this is that plugging in k = 10 gives us 10x + 20y = 10x + 10y, or 10y = 0. Similarly, plugging in k = 20 gives us 10x + 20y = 20x + 20y, or 10x = 0. Since x and y are positive, neither of these is possible.

It is not immediately clear at this point whether k should be between 10 and 20 (as in B, C, D) or above 20 (as in E). There are more answers between 10 and 20, so we can start there.

We can start with k = 15, which is in the middle of the three choices B-D. This gives us 10x + 20y = 15x + 15y, or 5y = 5x, or x = y. Since we are told that x < y, this cannot be true.

It may not be totally clear at this point whether you should move to k = 12 or k = 18. Since we know that x < y, we need to end up in a situation in which we have an equation ax = by where a > b.

It turns out this is the case with k = 18, though if you tried k = 12 first, you would quickly realize that this is wrong and try k = 18.

k = 12 gives us 10x + 20y = 12x + 12y, or 8y = 2x, or x = 4y. Since we are told that x < y, and since they are both positive, this cannot be true.

k = 18 gives us 10x + 20y = 18x + 18y, or 2y = 8x, or y = 4x. This meets the x < y condition, so it is the correct answer.

The algebraic approach taken in the guide is not intuitive, and rather difficult. Even if you just plugged in answers A-E in order for k, you would be more likely to get to the correct answer faster and with less confusion.

Of course, if you started with B and D, as is typically advised when plugging in answer choices, you would have gotten there even faster, since the answer happens to be D.


230.
Here, they start with x * 2^-17 = (...)

As they move from step 1 to step 2, they DIVIDE by 2^-17.
This puts 2^-17 in the denominator.

In the next step, they simplify the expression. First, they use the definition of negative exponents, which says that 2^-17 = [1/(2^17)].

In general, when we divide by 1/n, it is the same as multiplying by n. In this case, dividing by [1/(2^17)] is the same as multiplying by (2^17), so having 2^-17 in the denominator is the same as having 2^17 in the numerator.

Therefore, when we simplify, we get (...) * 2^17. From here, they distribute 2^17 across the four terms in the numerator, and then apply exponent rules to simplify each term.
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Re: Two questions from OG 12 Edition

by Ian Stewart » Mon Apr 06, 2009 6:03 pm
gmat_mba wrote:I was wondering if there is a typo in Problem Solving Answer Explanations for questions 148 and 230. Please remember these questions are from OG 12 Edition.

In Q 148: How did they get "y"in (10(x+"y") + 10y) / (x +y)? Please note this is the third step in the answer.
In Q 230: The answer explanation said 2^17 instead of 2^-17. Is this right?
There are a few approaches one can take to Q148, and it's often useful to see a variety of solutions. I've posted this one elsewhere:

Say you were asked the following question: "There are x men and y women at a certain company. The average wage of the men is $10 per hour, and the average wage of the women is $20 per hour. If k is the average wage of all employees at the company, what is k?" Here, you have x+y employees in total, the men combined earn 10x in total, and the women combined earn 20y in total, so

k = (10x + 20y)/(x + y)

That's the equation in the question -- it's just a weighted average. If the average of one group is 10, and the other is 20, the weighted average must be between 10 and 20, and if y > x (we have more women than men) then the average must be closer to 20 than to 10, so 18 is the only possible value of k among the answer choices.

Alternatively, you could do the question algebraically, as follows. Cross multiply to get:

10x + 20y = kx + ky
20y - ky = kx - 10x
y(20 - k) = x(k - 10)
y/x = (k - 10)/(20 - k)

We know y > x > 0, so the left side is bigger than 1. Notice that the right side is negative unless 10 <= k < 20, and that the numerator will only be larger than the denominator if k > 15. So 15 < k < 20.

___________________


As for Q230, I don't much like the OG's solution, because it's not easy to learn something from it that could be applied to other similar questions. There is a general principle that is useful to notice here - when adding or subtracting exponential expressions with the same base, you almost always want to factor out the term with the smallest power. For example, most often when you see something like x^3 + x^2, you'll normally want to factor out x^2 to get x^2(x + 1). The principle applies here as well; in the numerator, 2^(-17) is the smallest power. So:

[2^(-14) + 2^(-15) + 2^(-16) + 2^(-17)]/5 = 2^(-17)*[2^3 + 2^2 + 2^1 + 1]/5
= (2^(-17))*(15)/5
= 3*(2^(-17))

So the answer is 3.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by GHong14 » Tue Jan 25, 2011 2:32 pm
Thanks you guys this totally makes sense now. A perfect example where using the answer choices is helpful to get you what you want!

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by OneTwoThreeFour » Thu Jan 27, 2011 11:37 am
I used another approach to tackle this problem. Since it is always best to factor out the expression before taking it straight on, what I did it was 2 ^ -14 (1 + 2 ^ -1 + 2 ^ -2 + 2 ^ -3) / 5 / 2 ^-17.
Thus: 2 ^ -14 (1 + 1/2 + 1/4 + 1/8) / 5 / 2 ^-17.
Since 1 + 7/8 is roughly 2 then, the expression essentially becomes:

2 ^ -13 / 5 / 2 ^ -17.

The final answer is 3.2 which is closest to 3.