in terms of k and m

sanju09 wrote: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

sanju09 wrote: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)

Let the initial price per share be x and the initial earnings per share be y. After the increase, price per share became x(1 + k/100) and earnings per share became y(1 + m/100). The initial ratio of price per share to earnings per share was x/y and it became [x(1 + k/100)]/[y(1 + m/100)]. Let’s apply the percent increase formula:

Percent increase = 100 * (new - old)/old

Percent increase = 100 * [[x(1 + k/100)]/[y(1 + m/100)] - x/y]/[x/y]

Let’s factor out x/y from the numerator and cancel with the denominator:

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

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

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

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

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

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

Answer: D