Last year the price per share of Stock X increased by k percent and

My question to the helpful “experts” out there is how do you know when you will just plug in #s as opposed to going thru the problem the “normal” way? That decision alone takes me 1/2 a valuable minute to make :/

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Sirjesq
There is no yardstick to ensure when to take what decision.

BUT when you start feeling inconvenience with so many algebraic expressions then start substituting.

If the algebraic expression can be simplified without much inconvenience then simplify it first.

The simplification of algebraic functions is greatly helpful in DS questions specifically.

I am not sure how helpful this response is. But it was just an effort to explain what goes in head.

Video solution from Quant Reasoning starts at 15:28 here:

Let's try substitution method as the Question and answers are based on variables k and m.

Assume that price per share of stock X= 100 $
Earning per share of stock X = 100$
The ratio of price per share to earnings per share= 100/100 = 1

Its given that k > m
Let's assume k = 100 % and n=50 %. You can assume any values for K and M (k>m) .Also please make sure that calculation wise, it will be easy for you .

So, the price per share of Stock X is increased by 100 % and the earnings per share of Stock X is increased by 50 %

New price per share of Stock X = 100 + 100 % of 100 = 200 $
New Earning per share of stock X= 100 + 50 % of 100 = 100 + 50 = 150 $

The new ratio of price per share to earnings per share = 200/150 = 4/3
Percentage increase in the ratio of price per share to earnings per share = (value change/initial value )* 100 =
\((\frac{4}{3} -1)/1\)*100
= (1/3)* 100 = 33.33 %


Now we will substitute k= 100 and m= 50 in the options and see which one will give you 33.33 %.
Only option D would give 33.33 %.
\(\frac{100(k–m)}{(100+m)}= \frac{100*50}{150} = 100/3 = 33.33 %\)

Other options can be easily eliminated and D is the correct answer.

Thanks,
Clifin J Francis

BrentGMATPrepNow
Thank you so much for your explanation. To clarify, we need to know k>m for picking numbers, but not for algebra in this case? Would you have needed to pick a different set of numbers if you had two answer choices that yielded the same number? Thank you again.

The whole k > m is meant to ensure that are there actually is an increase in the ratio of price per share to earnings per share.

Yes, if you end up with two answer choices yielding the same number, then you must test another value (or pair of values in this case) in order to eliminate one of the remaining answer choices.
More here:

Hi, can you share links to more questions like these?
I am making errors and I think fumbling in keeping neat calculations. Do you have any specific suggestion? Right now I feel only practicing more questions like these and making it a habit to keep clean notes will help
Looking forward to your response. Thanks!

Given: Last year the price per share of Stock X increased by k percent and the earnings per share of Stock X increased by m percent, where k is greater than m.

Asked: By what percent did the ratio of price per share to earnings per share increase, in terms of k and m?

Let the price of the share be P & earnings per share be E.

Last Year: -
Price per share = P
Earning per share = E
Price per share to earnings per share = P/E

This Year: -
Price per share = P(1+k%)
Earning per share = E(1+m%)
Price per share to earnings per share = P(1+k%)/E(1+m%)

Percentage increase in Price per share to earnings per share = 100%{P(1+k%)/E(1+m%) - P/E}/(P/E) = {(1+k%)/(1+m%) - 1}*100% = {(100+k)/(100+m) - 1}*100% = 100(k-m)/(100+m) %

IMO D

To determine the percent increase in the ratio of price per share to earnings per share, we need to compare the changes in these two values.

Let's assume the initial price per share is P and the initial earnings per share is E.

After the increase of k percent, the new price per share becomes P + (k/100)P = P(1 + k/100).

After the increase of m percent, the new earnings per share becomes E + (m/100)E = E(1 + m/100).

The ratio of the new price per share to the new earnings per share is (P(1 + k/100))/(E(1 + m/100)).

To find the percent increase, we calculate the difference between the new ratio and the initial ratio, divide it by the initial ratio, and multiply by 100%:

Percent increase = [(New ratio - Initial ratio) / Initial ratio] × 100%

Let's calculate the new ratio:

New ratio = (P(1 + k/100))/(E(1 + m/100))

Now, let's calculate the percent increase:

Percent increase = [(P(1 + k/100))/(E(1 + m/100)) - (P/E)] / (P/E) × 100% = [(P(1 + k/100))/(E(1 + m/100)) - (P/E)] / (P/E) × 100% = [(P(1 + k/100)) - (E(1 + m/100))] / (E(1 + m/100)) × (E/P) × 100% = [(P + (k/100)P) - (E + (m/100)E)] / (E(1 + m/100)) × (E/P) × 100% = [(k/100)P - (m/100)E] / (E(1 + m/100)) × (E/P) × 100% = [(k/100) - (m/100)(E/P)] / (1 + m/100) × 100%

Since the given question asks for the percent increase in terms of k and m, we can express the ratio of E/P as 1.

Percent increase = [(k/100) - (m/100)] / (1 + m/100) × 100% = [(k - m)/(100 + m)] × 100%

Therefore, the percent increase in the ratio of price per share to earnings per share, in terms of k and m, is 100(k - m)/(100 + m)

Best way to solve under 1.5m is to just think strategically.
1. think of x/y which is increased by k and m respectively.
2. since we don't know whether x >y or not, we can be sure the answer is not k/m or k-m(has to have *100 element). Eliminate A,B.
3. change percentage formula is just increase/ base. All three options have k-m (which denotes the increase). Base should have m (since its related to earning's increase percentage).­

Can somebody please explain what I’m doing wrong? I don’t see how the answer can be D based on my math. Where am I going wrong in my screenshot I don’t understand how this problem works. I plugged my own numbers in and when I plugged them back into the answer choices, I cannot arrive at my answer choice. It’s very frustrating.

Hey Bunuel,

after reading your clarification, I still fail to grasp "reduce by \(\frac{x}{y}\)" and turning into
\(\frac{\frac{x(100+k)}{y(100+m)}-\frac{x}{y}}{\frac{x}{y}}100=\frac{\frac{100+k}{100+m}-1}{1}100\)




I appreciate the assistance on this.

I'm not sure what else I can add to what I've already written. Maybe you could try to pinpoint exactly what is unclear to you?

I think I'm stuck on "reducing" by x/y, as in how we can eliminate it, so it results in \(\frac{\frac{100+k}{100+m}-1}{1}\)

I hope this isn't too silly of a question..

Factor out x/y:

\(\frac{\frac{x(100+k)}{y(100+m)}-\frac{x}{y}}{\frac{x}{y}}100 =\)

\(=\frac{ \frac{x}{y} (\frac{(100+k)}{(100+m)}-1)}{\frac{x}{y}}100 =\)

\( =\frac{\frac{100+k}{100+m}-1}{1}100= \)

Ah, perfect. Thank you.

@KarishmaB and Bunuel

I tried to solve by plugging in numbers, but I got stuck at one point.
I considered Share Price to be 200 and Earnings per share to be 100. Thus their ratio is 2:1.
I considered k = 100 and m = 50.
Therefore the new price : new earnings = 400 : 150 = 8:3
The increase in the ratio = 8/3 - 2 = 2/3 %
But I am unable to get 2/3 as an answer after plugging in values for any of the options. Am I doing something wrong here?

Is it necessary to consider the initial ratio as 1:1?

Could you please guide me here.