The annual rent collected by a corporation from a certain building was

Question

The annual rent collected by a corporation from a certain building was x percent more in 1998 than in 1997 and y percent less in 1999 than in 1998. Was the annual rent collected by the corporation from the building more in 1999 than in 1997?

(1) $x > y$

(2) $\frac{xy}{100} < x-y$

(1)  x > y

(2)  \frac{xy}{100} < x-y

Bunuel · Accepted Answer

The annual rent collected by a corporation from a certain building was x percent more in 1998 than in 1997 and y percent less in 1999 than in 1998. Was the annual rent collected by the corporation from the building more in 1999 than in 1997?

Given:

Question:

(1) $x>y$,

Based on this information we cannot conclude whether $x-y>\frac{xy}{100}$. Not sufficient.
(2) $\frac{xy}{100} < x -y$.

This, directly gives an YES answer to the question. Sufficient.
Answer: B.

KarishmaB · Answer

This is a question based on successive percentage changes i.e. a number changes by a certain factor, then it changes again by some other factor and so on. Take 100 for example. I first increase it by 10% so it becomes 110. I then decrease 110 by 20% so it becomes 88.
I personally favor multiplying the factor of change together e.g.
$100 * (1 + \frac{10}{100}) * (1 - \frac{20}{100})$ = 88

but a lot of my students like to use the simple formula of two successive percentage changes which is: $a + b + \frac{ab}{100}$
(You can derive it very easily. Let me know if you face a problem)

If a number is changed by a% and then by b%, its overall percentage change is as given by the formula.
Using it in the example above, a = 10, b = -20 (since it is a decrease)
$a + b + \frac{ab}{100}$ = $10 - 20 - \frac{10*20}{100}$ = -12%
So overall change will be of -12%. 100 becomes 88.

In our question above, since we have two successive percentage changes of x% and y% (which is a decrease) so the formula gives us $x - y - \frac{xy}{100}$ is the overall change. The question is, whether this change is positive i.e. whether $x - y - \frac{xy}{100}$ > 0 or
$x - y > \frac{xy}{100}$?
Since statement 2 tells us that $x - y > \frac{xy}{100}$, it is sufficient.

And yes, it helps to be clear about exactly what is asked before you move on to the statements.

Here is a post on successive percentage changes: https://anaprep.com/arithmetic-successi ... e-changes/

Bunuel · Answer

r 0<-\frac{y}{100}+\frac{x}{100}-\frac{xy}{10000} , multiplu by 100 both sides --> 0<-y+x-\frac{xy}{100} , rearrange --> x-y>\frac{xy}{100} . Hope it's clear.

KarishmaB · Answer

You can use the formula in the question in a very straightforward manner...
Rent charged in 1998 was x% more
Rent charged in 1999 was y % less
So overall % change in rent charged was x - y - xy/100 (since y is a decrease use (-y) in place of y)
Question: Was rent collected in 1999 > 1997 i.e. was overall percentage of change positive? 
Was x - y - xy/100 > 0 ?

(1) x > y
Not enough info

(2) xy/100 *< x – y
which is basically just x - y - xy/100 > 0 when you re-arrange. 
So this statement gives you 'Yes' immediately. Sufficient.

Though, make sure that you know how the formula was derived and the basic concept behind it... Its good to know short cut formulas but they are not applicable everywhere... in some questions you might need to use ingenuity...

mcelroytutoring · Answer

I do not recommend using algebra to evaluate condition #2--it's too complicated for most. Here is a visual that should help, showing you that you can evaluate condition #2 by plugging in to determine the "boundary line" (the value that separates the "Yes" answers from the "No" answers).

Attachments

Screen Shot 2017-01-17 at 6.13.59 PM.png [ 131.74 KiB | Viewed 26126 times ]

JeffTargetTestPrep · Answer

We are given that the rent collected in a building was x percent more in 1998 than it was in 1997 and y percent less in 1999 than it was in was in 1998. Let’s start by defining some variables. a = the annual rent collected in 1997 b = the annual rent collected in 1998 c = the annual rent collected in 1999 We can now create the following equations, using the "percent greater than" and "percent less than" formulas: b = a c = b We need to determine whether the annual rent collected by the corporation was more in 1999 than in 1997. Thus, we need to determine: Is c > a? Since b = a and c = b, that means c = a. Now we can rephrase the question as: Is a > a? Notice if we divide the entire inequality by a, we have: Is > 1? Is (100-y)(100+x)/10,000 > 1? Is (100+x)(100-y) > 10,000 ? Is 10,000 – 100y + 100x – xy > 10,000 ? Is -100y + 100x – xy > 0 ? Is 100 x – 100y > xy ? Is 100(x – y) > xy ? Statement One Alone: x > y Knowing only that x is greater than y is not enough to determine whether 100(x – y) > xy. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D. Statement Two Alone: (xy/100) < (x-y) Multiplying both sides of the inequality by 100, we have: xy <100(x – y) xy < 100(x – y) is exactly the same as saying 100(x – y) > xy. Statement two alone is sufficient to answer the question. Answer: B

avigutman · Answer

Video solution from Quant Reasoning starts at 23:03 here:
 https://www.youtube.com/watch?v=uy6AaZHs1YI

testtakerstrategy · Answer

VeritasKarishma  -- why can we cancel out the r? We dont necessarily know it is not equal to zero.

KarishmaB · Answer

r is the rent collected in 1997. It cannot be 0. If it were, how would you collect x% more in 1998? x% of 0 is still 0 so rent collected in 1998 will again be 0 + 0 = 0 (which is not more than rent collected in 1997).
In such real world questions, the base is not 0.

PGTLrowanhand · Answer

Hi GMATters,

Here is my video solution to this question:

https://www.youtube.com/watch?v=4S_-B5X4flg

Best,

Rowan

ADisHere · Answer

Hi, i very well understood your algebraic method. it is very methodical. but I cant logically conclude why d isn't the answer.

See what I thought for (a) is that as x>y then the x% will be greater than y%. so obviously as the increase is greater than decrease, the answer will be greater than the original.
i know my thinking has a major flaw. but I want to get it cleared here itself. please help!!

Bunuel · Answer

The point is that the increase, x%, and the decrease, y%, are applied to different amounts.

For example, if the rent in 1997 was $100, and x = 50% and y = 10%, then in 1998 the rent would be $150, and in 1999 $135, giving a YES answer to the question ($135 > $100).

However, if the rent in 1997 was $100, and x = 50% and y = 40%, then in 1998 the rent would be $150, and in 1999 $90, giving a NO answer to the question ($90 < $100).

Hope it's clear.

ADisHere · Answer

Thanks a lot, Bunnel, Your Highness!!
I cant thank you enough for your beautiful and elegant explanations!

You truly are a blessing to the Gmat community!

GGGMAT2 · Answer

KarishmaB Thanks for another wonderful explanation post.

I was going through the link provided in your post i.e. https://anaprep.com/arithmetic-successi ... e-changes/

The link has an example -
" Six months back, the cost of an air ticket from Detroit to San Francisco was €400. Four months back, the fares increased by 12.5%. Last month, the fares increased by 25% and yesterday, the airlines again increased the fares by 11.11%."

Here specific percent increases or decreases such as increase of 12.5%, increase of 25%, and increase of 11.11%, are converted into neat ratios -

.

Is there another link that would explain how to quickly convert percentage increase or decrease to ratios. I know 25 increase is 5/4 since its commonly used, but it would really help know how 12.5% increase gets converted to 9/8 and 11.11% increase gets converted to 10/9. Thanks!

KarishmaB · Answer

Check out my Percentages video on YouTube. Explains all this and more.
Percentages Concept Video:  https://youtu.be/HxnsYI1Rws8

GGGMAT2 · Answer

Thanks   KarishmaB   . I just went through the suggested video and it was really helpful.

agrasan · Answer

Hi experts  Bunuel   KarishmaB

Why S(1) is not an answer?

My logic is  xy/10000 ~ 0  if we look at the equation: 
 
 Is 1< 1− y/100 + x/100 − xy/100001 <1− y/100 + x/100 −xy/10000 ? 
 
Please let me know if it is wrong to assume xy/10000 close to 0.

Bunuel · Answer

Yes, it is wrong to ignore xy/10000. In my post above, I give two examples where Statement (1) leads to different answers:

For example, if the rent in 1997 was $100, and x = 50% and y = 10%, then in 1998 the rent would be $150, and in 1999 $135, giving a YES answer to the question ($135 > $100).

However, if the rent in 1997 was $100, and x = 50% and y = 40%, then in 1998 the rent would be $150, and in 1999 $90, giving a NO answer to the question ($90 < $100).

Also, the question essentially boils down to simpler form: "is  x-y>\frac{xy}{100} ?"

Hope it's clear.

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The annual rent collected by a corporation from a certain building was

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