Problem Solving

Official Guide

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?

$$\text{A. } \frac{k}{m}$$
$$\text{B. } (k-m)$$
$$\text{C. } \frac{100(k-m)}{(100+k)}$$
$$\text{D. } \frac{100(k-m)}{(100+m)}$$
$$\text{E. } \frac{100(k-m)}{(100+k+m)}$$

OA D.

AAPL wrote:Official Guide

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?

$$\text{A. } \frac{k}{m}$$
$$\text{B. } (k-m)$$
$$\text{C. } \frac{100(k-m)}{(100+k)}$$
$$\text{D. } \frac{100(k-m)}{(100+m)}$$
$$\text{E. } \frac{100(k-m)}{(100+k+m)}$$

OA D.

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)(1) = 3.
Since the earnings increase by m=100%, the new earnings = 100 + (100/100)(1) = 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 = 100.
Original ratio of price to earnings = 100/100 = 1.

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