An estimate of an actual data value has an error of p percent if p = |

It's a tough Question because of a lot of data input

Video solution is attached here.

Answer: Option A


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Using the same variables defined in the OA:

\(E_I\) = Estimated Total Income
\(E_T\) = Estimated Tutoring Income
\(A_I\) = Actual Total Income
\(A_T\) = Actual Tutoring Income

The problem tells us that in comparing estimates to actual figures:

Total Income Error < 20%
Tutoring Income Error < 20%

The question is:

Is \(\frac{A_T}{A_I}\leq{0.45}\)?

(1) To make this easy, set \(E_I\) = 100. Then according to statement (1), \(E_T\) = 0.3\(E_I\) = 30, and therefore:

80 < \(A_I\) < 120 since the error is at most +/- 20%
24 < \(A_T\) < 36 since the error is at most +/- 20%

\(\frac{A_T}{A_I}\) is maximized when \(A_T\) is maximized and \(A_I\) is minimized, so the most \(\frac{A_T}{A_I}\) could be is:

\(\frac{A_T}{A_I}\) = \(\frac{36}{80}\) = 0.45 SUFFICIENT (DEFINITIVELY YES)

(2) We don't know anything about how much of Emma's total income was from tutoring. INSUFFICIENT

Answer A

Veritas Karishma,Bunuel,

Please help to solve this one. Unable to do within 2 mins.

Hi Gmat20201

A great misconception among the GMAT aspirants is that every question should be solved in 2 minutes while this is totally absurd notion.

Please understand that while we need to maintain the average time per question as 2 minutes, it doesn't expect us to give every question 2 mins and get successful in solving.

While Some questions take less than a minute to solve, there are some questions that very well deserve even 3 minutes or maybe more (in test situations which are often stressful). This problem is exactly one of the latter type problems which deserve more than 2 minutes and hence one should be rational to understand this fact and not push himself/herself to solve it in 2 minutes.

Happy learning!!!

Under one min solution:

(A)
Simple assume numbers

Total Income (TI)= 100
=> Max TI (20% More)= 120 and Min TI (20% less) = 80

Tuition Income = 100*30% = 30
=> Max Tuition Income (20% More)= 36 and Min Tuition Income (20% less) = 24

Case when tuition income is the highest % of total income. When Total income is at min at 80 and tuition income is at max at 36.
Calculate: Tuition inc (Max)/ Total Income (Min) = 45%. (At most)
SUFFICIENT

(B)
No information about Tuition Income
INSUFFICIENT

Hi,

This question's got a lot of fluff, but it's essentially a really easy question.
No variables needed :

So the formula given is just a regular formula that you already know :

Statement 1:

Let her estimated Total Income be 100 , and her estimated tut income be 30.

Now, since we have 20% error max,
100 can fluctuate between (80,120) and 30 can fluctuate between (24,36)

Now we want to make the estimated income a higher percentage of the actual income to test 1, basically, if we get anything above 45% we can ditch statement 1.

So let's try making the numerator as big as possible while keeping the denominator small.
Let's pick 80 for total and 36 for estimated.

36/80 * 100 = 45 %

But we're not allowed to pick 120 or 36 because they're equal to 20 percent and prompt says we gotta pick less than 20 %.

So let's say we picked 35/79 , question is, is (35/79) * 100 bigger than 45% or smaller?
Don't calculate.

For any proper fraction [less than 1] like for eg 1/2, if you add a certain value to numerator and denominator it gets bigger.

so for eg if we add one to num and denom we get 1+1/2+1 which is 2/3 and 2/3 > 1/2
So similarly, 36/80 has to be bigger than 35/79 , hence 35/79 * 100 is less than 45 percent

I'm so confused.
Both OG solution and the video from GMATinsight say that 0.8E<A<1.2E

But according to VeritasKarishma,


it's 0.8A<E<1.2A.

Are they both correct?

eleanorlee2116

Given:
p = 100|e - a|/a

If p = 20,

20/100 = |e - a|/a

a/5 = |e - a|

Two cases:
e - a = a/5
e = 6a/5
or
e - a = -a/5
e = 4a/5

So 4a/5 < e < 6a/5
0.8a < e < 1.2a

The estimate must lie between '20% less than actual' and '20% more than actual'. The base of the percentage must be the "actual".

Share the exact official solution with me and I can explain it to you. Very unlikely that it is incorrect.

A bit confused here.
Since the actual is between "(5/6)*Estimate" and "(5/4)*Estimate", so shouldn't we be using a fraction lower than 5/4 for calculating the tutoring income? Similarly, shouldn't we be using a fraction greater than 5/6 for calculating the total income?

jaggisrishti - It doesn't matter when we are not required to work with integers.
It is easy to work at the extreme data points. I would much rather work at 20% (which I can say is equivalent to 19.99999999999%) than at 19% even though the question says less than 20%.
I get that tutoring income could be at most 375. This just tells me that tutoring income could take a value 374.999999 and total income could take a value 833.00000000001.
Since 375/833 gives me 45%, 374.9999999/833.0000001 will certainly give me a little less than 45%.

I only need to keep an eye on which direction my values are in from the extreme position.

I strongly believe that the community needs the wizard IanStewart to post "his solution" to this question.

One of the unfortunate consequences of the kudos system on GMAT Club is that solutions that are incorrect can rise to the top of a thread, and then the sheer number of kudos can lead future readers to think the solution is correct. That's true in this thread -- both of the top-voted posts from GMATInsight and "SimplyBrilliant" are mathematically wrong (unless they edit their posts after reading this). Skimming the thread, the two solutions that are correct are, as you'd expect, those from KarishmaB and from nick1816.

Here the percent error of an estimate is taken as a percentage of the actual value. So if an actual value is 100, and an estimate of that value has less than a 20% error, the estimate is between 80 and 120. The two incorrect solutions at the top of this thread do this backwards -- they take the error as a percentage of the estimate, so they incorrectly conclude that if an estimate is 100, and has less than 20% error, the actual value is between 80 and 120. That's not right -- the actual value in this case is between 83 1/3 and 125. So those solutions make two mistakes, and just get lucky that the two errors compensate for each other and produce the right answer.

Anyway, my solution, which is essentially the same as Karishma's: first, the answer here can really only be A or E, because there's no reason in a pure percent question like this that we'd care about the $40,000 number in Statement 2.

If an actual value is V, and an estimate has less than 20% error, then the estimate is between 0.8V and 1.2V, so it is between (4/5)V and (6/5)V. So in the case where the estimate is as low as possible, the true value is 5/4 of the estimate (because (5/4)(4/5)V = V), and similarly when the estimate is as high as possible, the true value is 5/6 of the estimate.

Using Statement 1 alone, we want to know just how big a percentage of her income her tutoring could represent. So we want to maximize her tutoring income, and minimize her total income. If her estimated total income was T, then, from the above, her total income was greater than (5/6)T. Her estimated tutoring income was 30% of her estimated income, so was 3T/10, and her actual tutoring income was no more than (5/4)(3T/10) = 3T/8. So in the limit case, where she made as much as possible from tutoring as a percentage of her overall income, she made

(3T/8) / (5T/6) = 9T/20 = 0.45T

or 45% of her income from tutoring. Since we have inequalities, but we assumed our values equaled the boundary values, we can actually be sure that less than 45% of her income came from tutoring, and Statement 1 is sufficient.

It appears after I wrote my post that a GMAT Club moderator demoted the two incorrect solutions so they are no longer at the top of the thread.

It would be considerate if mods could post a quick explanation when they do things like that, because without any explanation, it could appear to a reader that I made a mistake i didn't make when describing the contents of this thread (and when I say the top post in this thread is wrong, that's potentially unfair to KarishmaB, who is now deservedly the top post in this thread).

avigutman

If you have time, I would be so appreciative to learn how you would approach this problem. The approaches on this form seem a bit data intensive... curious if there is an alternative way

My immediate reaction when glancing at the question and the statements was to eliminate most of the answer choices (BCD), just as IanStewart correctly observed above. That is a very important DS observation/strategy.
Between A and E, I'm afraid I can't think of a non-computational approach to picking the answer, so if you can't solve the way Ian did, the correct approach is to take a 50-50 guess between A and E, and move on, woohoo921!

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did not get this part \(\frac{5}{6}e_l < a_l < \frac{5}{4}e_l\)...........(1)
, how we inferred this

akt715 - Did you understand till the previous step?
\(\frac{4}{5}a_l < e_l <\frac{ 6}{5}a_l\)

If yes, then note that the next step is derived from there. Take one inequality at a time:
\(\frac{4}{5}a_l < e_l\)
\(a_l < \frac{5}{4}e_l\)

and

\(e_l <\frac{ 6}{5}a_l\)
\(\frac{ 5}{6}e_l <a_l\)

So we get:
\(\frac{ 5}{6}e_l <a_l < \frac{5}{4}e_l\)