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

[ideree]

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

(A) k% / m
(B) (k - m)%
(C) 100(km) % / 100+k
(D) 100(k − m) % / 100+m
(E) 100(k − m) % / 100+k +m

Please help me to find better explanation <i class="em em-pray"></i>

[deloitte247]

Let the original price = $$x$$
Original earnings = y
original ratio price per earning = $$\frac{x}{y}$$
Increased price = $$x\left(1+\frac{k}{100}\right)$$ = $$\frac{x\left(100\ +k\right)}{100}$$
Increased earnings = $$y\left(1+\frac{m}{1000}\right)$$ = $$\frac{y\left(100+m\right)}{100}$$
$$new ratio price per earning = \frac{x\left(100+k\right)}{y\left(100+m\right)}$$
$$\triangle\%\ =\ \frac{Change}{Original}\cdot100$$
$$=\frac{\left[\frac{x\left(100+k\right)}{y\left(100+m\right)}-\frac{x}{y}\right]}{\frac{x}{y}}\cdot100$$
$$=\left[\frac{\left(100+k\right)}{100+m}-1\right]\cdot100$$
$$=\left[\frac{\left(100+k-100-m\right)}{100+m}\right]\cdot100$$
$$=\frac{100\left(k-m\right)}{100+m}$$
Answer = Option D

[GMATGuruNY]

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

(A) k% / m
(B) (k - m)%
(C) 100(km) % / 100+k
(D) 100(k − m) % / 100+m
(E) 100(k − m) % / 100+k +m

Original ratio:
Let the original price = 1 and the original earnings = 1.
Resulting ratio:
price/earnings = 1/1 = 1.

New ratio:
Let k=200 and m=100.
Since the price increases by k=200%, the new price = 1 + (200/100)(100) = 3.
Since the earnings increase by m=100%, the new earnings = 1 + (100/100) = 2.
Resulting ratio:
(new price)/(new earnings) = 3/2 = 1.5.

Since the old ratio = 1 and the new ratio = 1.5, the ratio increases by 50%. This is our target.
Now plug k=200 and m=100 into the answers to see which yields our target of 50.
Only D works:
100(k-m)/(100+m) = (100)(200-100) / (100+100) = 50.

The correct answer is D.

An alternate approach is to combine plugging in values with a bit of algebra.

Old ratio:
Let the original price = 100 and the original earnings = 1--.
Original ratio of price to earnings = 100/100 = 1.

New ratio:
Price increased by k% = 100 + (k/100)(100) = 1 + k.
Earnings increased by m% = 1 + (m/100)(100) = 100 + m.
New ratio = (100+k)/(100+m).

Difference between the ratios:
(100+k)/(100+m) - 1 = [(100+k) - (100+m)] / (100+m) = (k-m)/(100+m).

Percent change in the ratios = (difference between the ratios)/(original ratio) * 100:
[(k-m)/(100+m)] / 1 * 100 = [100(k-m)] / (100+m).

The correct answer is D.

[Brent@GMATPrepNow]

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

A) k/m%
B) (k-m)%
C) 100(k-m)/100+k %
D) 100(k-m)/100+m %
E) 100(k-m)/100+k+m %

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: D

Cheers,
Brent