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

First note that p = 100*|e - a|/a

This is nothing but formula of percentage difference.
If I ask - "Estimate differs from actual by what percentage," how will you calculate that? 

You will say (e - a)/a * 100.
That is what the given formula is except that we have an absolute value sign to ignore negatives.

So say the actual value is $100, I can say that if error percentage is 20%, the estimate value will lie between 80 and 120. 
Now we are given that tutoring income and total income error percentages are less than 20%. 


Estimate = Between "(6/5)*Actual" and "(4/5)*Actual"

This means Actual = Between "(5/6)*Estimate" and "(5/4)*Estimate"

Stmnt 1: Estimated tutoring income was 30% of estimated total income.

Say estimated total income was $1000, tutoring income was $300. 

To get maximum value of tutoring as a part of total, maximise tutoring income and minimise total income.
So tutoring income could be at most (5/4)*300 = 375
Total income will be at least (5/6)*1000 = 833
This gives us 375/833 * 100 = 45%

So actual income from tutoring is at most 45% of total actual income.
Sufficient.

(2) Emma's estimated total income last year was $40,000.
No clue about her tutoring income.
Not sufficient.

Answer (A)
_________________

Thanks Sir.

Emma's estimated total income last year = \(e_l\)

Emma's actual total income last year =\( a_l\)

Emma's estimated income from tutoring last year = \(e_t\)

Emma's actual income from tutoring last year = \(a_t\)

\(\frac{|e_l - a_l|}{a_l }< \frac{1}{5}\)

\(4a_l < 5e_l < 6a_l\)

or

\(\frac{4}{5}a_l < e_l <\frac{ 6}{5}a_l\)

or

\(\frac{5}{6}e_l < a_l < \frac{5}{4}e_l\)...........(1)

Similarly we get

\(\frac{5}{6}e_t < a_t < \frac{5}{4}e_t\)...........(2)


Statement 1-

\(e_t = \frac{3}{10} e_l\)........(3)

From (2) and (3)

\(\frac{5*3}{6*10}e_l < a_t < \frac{5*3}{4*10}e_l\)

\(\frac{1}{4}e_l < a_t < \frac{3}{8}e_l\)

Maximum possible value of \(\frac{a_t}{a_l}\) occurs when \(a_t\) is max and \(a_l\) is minimum.

\((\frac{a_t}{a_l})_{max} < [(\frac{3}{8}) e_l] ÷ [(\frac{5}{6}) e_l]\)

\((\frac{a_t}{a_l})_{max} < \frac{9}{20}\)

\((\frac{a_t}{a_l})_{max} < 0.45\) or \(45\)%

Sufficient

Statement 2-

e_l = 40000, it gives the range of \(a_l\). But we know nothing about \(a_t\).

Insufficient

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

Video solution is attached here.

Answer: Option A





ACCESS VIDEO SOLUTIONS OF ALL QUANT AdVANCED GUIDE PROBLEMS from my website https://www.GMATinsight.com in price section

A piece of advice whenever you see any lengthy questions, Always get the data in simpler form.(for eg. I use table)
Let's see info given

Income------------------Estimate---------Actual---------Error
Total income------------e1-----------------a1---------------(e1-a1)
Tutor income-------------e2----------------a2----------------(e2-a2)

As per question,Emma's estimate for her total income last year had an error of less than 20 percent

Thus (e1-a1)/a1<1/5 ( 20%)

Simplifying above equation we get 5e1<2a1 and 5e2<2a2

Now take option A
e2=0.3e1

If we substitute it highlighted equation.
We will get ratio of a1/a2.

Hence A is sufficient.

Option B is not sufficient as it doesn't give any info on income from tutoring.

hope it helps

Harsh2111s

Though you got the correct answer, you solution is not correct.

For example

If x<2 and y<3

Can you find the upper bound of y/x ?

Answer is no (Why?).

y/x tends to infinity, if y is a very large number and x is closer to 2.

To find the upper bound of y/x, you must know upper bound of y and lower for x.

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!!!

Lets take the variables:

Ai= Actual total income
At= Actual income from tutoring
Ei= Estimated total income
Et= Estimated total income from tutoring.

Lets say Ei= 100x ,
so, Ai can be 80x or 120x (considering +/- 20%)
from statement (1) we know Et=0.30*Ei = 30x
->so, At can be 0.24x or 0.36x (+/- 20% error margin)

From the qns ( actual income from tutoring last year at most 45 percent of her actual total income last year)
To do this, we need to minimize the Ai and Maximize At

Now to maximize At/Ai I will take the the minimum value of Ai (80x) (please note the value is actually more than
80x, since error percentage is less than 20%) and max value of At (0.36x) ( real value will be less than 0.36x )

So ((0.36x )/(0.80 x))*100 = 45 %,
But from above we know the At is slightly below 0.36x and Ai slightly above 0.80x, At/Ai will be below 45%


Option 2 on a standalone basis provides no info to answer the question asked.

therefore Answer is A

It becomes easy if we have an intuition of what the formula given stands for.

When we have percentages, for example in case of cost price & sell price, the profit in between CP & SP is our profit as a % of CP.

Therefore, if we understand the formula's logic, its easy to go further in acceptable time period.

I like to draw visual diagrams for % question so I can visually see what's going on.

The formula means the following :

Actual total income ------------(less than 20% difference)------------------- Estimated total income

AND....

Actual tutor income ------------(less than 20% difference)------------------- Estimated tutor income

(1) This says that Etutor = 0.3 Etotal.
So I assume Etotal as 100. And Etutor as 30.

Now, from our above logic of %, we can say that :

Actual total income ------------(less than 20% difference)------------------- 100

AND....

Actual tutor income ------------(less than 20% difference)------------------- 30

Now, we just need to find the range of Actual total & Actual Tutor Incomes.
Actual Total Income will be either LESS THAN 100 or GREATER THAN 100 (by 20%)

This range for Actual Total Income = (100/1.2) to (100/0.8)

Similarly, range for Actual tutor income = (30/1.2) to (30/0.8)

Now we need to check whether :

Actual tutor income <= 0.45 (Actual total income)

For this, because we need to find ATMOST (actual tutor income) we need to consider the maximum value for it --> which is (30/0.8).
And we need to minimize "Actual total income" because we want the upper limit to be as low as possible. ---> which is
(100/1.2)

The stem basically now translate into :
Is (30/0.8) <= 0.45 (100/1.2) --------------> And we get an answer YES.

Sufficient.

(2) this statement is clearly insufficient; There is no relative values provided. Just a concrete value for one variable. Notice that in statement 1 we have relative value (% of something) given, which makes the problem translate into ratios & helps us reach an answer.


Hope it helps.

Thanks & Regards,
Saakhi

Are your numbers correct here?
If estimated total is 1000, then isnt the error within 800 and 1200?
And 5/4 of 300 is 360.

360/800 is 45 %

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

THANK YOU for the brilliant solution
But a quick question. All through this solution, the formula for p was not used at all.
So was the formula for p, given in the question, nothing more than a trick to throw us off?

Regards
Sukriti

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.
_________________

In here although they say that the error is less than 20% , lets assume that the error is exactly 20% for ease of interpretation

now, assume
Et is estimated tutor income
At is actual tutor income
Ei is estimated tutor income
Ai is actual tutor income

Now the question tells us
Et +/- ( 0.2 Et) = At -----(i)
Ei +/- (0.2 Ei) = Ai -----(ii)
We want to figure out whether
At is atmost Ai +/- 0.45 Ai

1. First statement gives us

Et = 0.3 Ei, substitute Et in (i)

0.3 Ei +/- [ 0.2* 0.3 Ei] = At
0.3[ Ei +/- 0.2 Ei] =At
From (ii) we know Ei +/- (0.2 Ei) = Ai
hence, 0.3 Ai = At

Sufficient to conclude that At is atmost 45% of Ai hence A

2. Ei = 40000

With this we can estimate the value of Ai but it tells nothing about the relationship between Ei and Et or Ai ot At-> essentially we need something to connect the 2 equations

Insufficient