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.
Last year the price per share of Stock X increased by k
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Original ratio: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.
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.
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One approach is to plug in values.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 $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