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

The best way to solve such questions is to use Numbers. Let me show, how it works-
Pick Numbers which are easy to work with.
Lets assume that the Price of share before increase = 100
Assume k= 50%
So the price of share after increase = 150

Lets assume that the EPS before increase = 100
Assume m= 20%
So the EPS after increase = 120

Ratio of Price/EPS before increase = 100/100= 1
Ratio of Price/EPS after increase = 150/120= 1.25 ( a 25% percent increase, which we want to calculate & is obscured in terms of k & m)

To find the correct option, plug the value of k & m in options
A. k m % = 50 x 20 = 1000 = No
B. (k – m) % = 50-20 = 30 = No
C. [100(k – m)] / (100 + k) % = 100 (50-20)/(100+50) = 3000/150 = 20 -No
D. [100(k – m)] / (100 + m) % = 100 (50-20)/(100+20) = 3000/120 = 25 -Bingo
E. [100(k – m)] / (100 + k + m) % - Don't need to check this one.

Hope this help many.
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Sure: \(\frac{\frac{x(100+k)}{y(100+m)}-\frac{x}{y}}{\frac{x}{y}}100\) --> reduce by \(\frac{x}{y}\):

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

Hope it's clear.

Could you explain step-wise how you derived his one? I keep missing one step.

Thanks.

Responding to a pm: (super detailed steps)
When you have variables in the question and variables in the options, one method of getting to the answer is by assuming values for the variables. If a question says: x is greater than y/3 but less than y/2. Which of the following relations must hold?
(A) y < x < 2y
(B) 2y < 6x < 3y
...
...

One way of doing this is assuming some value for y. Say, y = 60. So x is greater than 20 but less than 30. Say x could be 25.
Not put y = 60 and x = 25 in the options:
(A) y < x < 2y -----> 60 < 25 < 2*120 ( does not hold)
(B) 2y < 6x < 3y ------> 2*60 < 6*25 < 3*60 (holds so could be the answer)

Mind you, you need to check every option to ensure that the relation holds for only one option. If it holds for more than one options, you will need to check for some other values too.

The point is that we assume values and then plug them in the options to see which relation works.

We can do this question using the same concept.
Last Year:
Price per share (PPS) increased by k%. Let's say k = 20. So if the price was 100, it increased by 20% to become 120 (New PPS).
Earnings per share increased by m%. Let's say m = 10 (m must be less than k). So if earnings per share was also 100, it increased by 10% to become 110 (New EPS).

By what percent did the ratio of price per share to earnings per share increase?

Ratio of PPS/EPS was 100/100 = 1
New ratio = New PPS/ New EPS = 120/110 = 1.0909

%Increase = [(1.0909 - 1)/1] * 100 = 9.09%

Now, in the options, put k = 20, m = 10
Only (D) will give you 9.09%. Let me show you calculations of some of the options.

B. (k – m) %
(20 - 10)% = 10% (Not the answer)

C. [100(k – m)] / (100 + k) %
[100(20 - 10)]/(100 + 20) % = 1000/120 % = 8.3% (Not the answer)

D. [100(k – m)] / (100 + m) %
[100(20 – 10)] / (100 + 10) % = 1000/11 % = 9.09% (Answer)

I think this type of problems lends itself well to easy number picking:

(P) share price= 100
(E) earnings=100
P/E = 1

Let k=200 and m=50

New P/E = 100*{(100+200)/100} / 100*{(100+50)/100} = 2 so increase of 100%

1) 200/50= 4 % wrong
2) 200-50= 150% wrong
3) 100(200-50)/(100+200)= (100*150)/300=50% wrong
4) 100(200-50)/(100+50)= (100*150)/150 = 100% Right
5) 100(200-50)/(100+200+50)= 100*150/350= wrong

--> Ans D

Earlier year share price = P
Earlier earnings = E
Ratio = P/E

Assume:
k = 50%
m = 25%

Therefore:
New share price = 1.5P
New earnings = 1.25E
New Ratio: 1.5P/1.25E = 150P/125E = 6P/5E = [1 + (1/5)]P/E
i.e. P/E ratio increased by 1/5 or 20%

Now, plug k = 50 and m = 25 in solutions. Only D yields 20%

Hence, Answer: D

First year:

Price = P
Earnings = E
Ratio#1 = P/E

Second year:

Price = P*K/100 + P
Earnings = E*M/100 + E
Ratio#2 = (P*K/100 + P)/(E*M/100 + E)

Percent increase in ratio = ((ratio#2 - ratio#1)/(ratio#1))*100
Percent increase in ratio = (((P*K/100 + P)/(E*M/100 + E) - P/E)/(P/E))*100

= ((K/100 + 1)/(M/100 + 1) - 1)*100

(K/100 + 1 - M/100 - 1)/(M/100 + 1)

=((K - M)/(M + 100))100

Karishma, in what cases would you advise to put in values rather than solve algebraically?

Let Original price =100
Let original earnings per share =100
Price/ earnings =1

New price = 100+k
New earnings per share= 100+m
price /earnings = (100+k)/(100+m)

Percent increase =((100+k)/(100+m) - 1) *100
= (k-m)*100/(100+m)

When the question doesn't give any concrete numbers and talks in terms of variables (the answer options would also be in terms of variables in that case), you can plug in values to get to the answer.
Percentage questions often provide an opportunity to plug in values too, say 100 for a base value.

How did you go from 12/11 to an increase of 100/11 percent?

According to original prices, PPS/EPS = 1
After the increase, PPS/EPS = 12/11

Percentage Increase in PPS/EPS \(=\frac{ (\frac{12}{11} - 1)}{1} * 100 = \frac{100}{11} %\)

how are you getting the -m in this step?

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

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

Let's use the INPUT-OUTPUT approach.

Another approach is to plug in values.

Let $100 be the original price per share of Stock X
Choose a "nice" value for k. How about k = 200
So, after a 200% increase, the new price per share = $300

Let $100 be the original earnings per share of Stock X
Choose a "nice" value for m. How about m = 100
So, after a 100% increase, the new earnings per share = $200

Original ratio of price/earnings = $100/$100 = 1
New ratio of price/earnings = $300/$200 = 1.5

By what percent did the ratio of price per share to earnings per share increase?
So, the percent increase (from 1 to 1.5) is 50%.
In other words, when k = 200 and m = 100, the ratio increases 50%

Now, plug in 200 for k, and 100 for m, and look for the answer choice that also yields 50%.

A. k/m = 200/100 = 2 (nope)

B. (k - m) = 200 - 100 = 100 (nope)

C. [100(k - m)] / (100 + k) = 10,000/300 = 33.333 (nope)

D. [100(k - m)] / (100 + m) = 10,000/200 = 50 GREAT!

E. [100(k - m)] / (100 + k + m) = 10,000/400 = 25 (nope)

Answer:

One approach is to plug in values.

Let $100 be the original price per share of Stock X
Choose a "nice" value for k. How about k = 200
So, after a 200% increase, the new price per share = $300

Let $100 be the original earnings per share of Stock X
Choose a "nice" value for m. How about m = 100
So, after a 100% increase, the new earnings per share = $200

Original ratio of price/earnings = $100/$100 = 1
New ratio of price/earnings = $300/$200 = 1.5

By what percent did the ratio of price per share to earnings per share increase?
So, the percent increase (from 1 to 1.5) is 50%.
In other words, when k = 200 and m = 100, the ratio increases 50%

Now, plug in 200 for k, and 100 for m, and look for the answer choice that also yields 50%.

A. k/m = 200/100 = 2 (nope)

B. (k - m) = 200 - 100 = 100 (nope)

C. [100(k - m)] / (100 + k) = 10,000/300 = 33.333 (nope)

D. [100(k - m)] / (100 + m) = 10,000/200 = 50 GREAT!

E. [100(k - m)] / (100 + k + m) = 10,000/400 = 25 (nope)

Answer:

hi

Would you please speak few more words about the expression "((100+k)/(100+m) - 1)" ....? especially about 1...? Is this something as such that "1", when subtracted from a fraction reveals the change, increase or decrease ....?

thanks in advance ...