Question

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. $ \dfrac{k}{m}\% $
B. $ \left( {k - m} \right)\% $
C. $ \dfrac{{100\left( {k - m} \right)}}{{100 + k}}\% $
D. $ \dfrac{{100\left( {k - m} \right)}}{{100 + m}}\% $
E. $ \dfrac{{100\left( {k - m} \right)}}{{100 + k + m}}\% $

Answer 1

Hint: o solve this question, we will first assume the values of the original price and original earning. Then we will find the ratio of the original price and earning. As the percentage change of earning and price are given in the question, we will find the new price and earning. Then we will find the new ratio of price and earning. After that, we find the difference between the new and original ratio. To find the percentage change, we will find the percentage of the difference between the ratio with respect to the original ratio.

Complete step-by-step answer:
We will assume that the original price of stock $ X $ was 100 and we will also assume the original earnings were 100.
The ratio of original price and original earning will be 1 as both are the same.
As given in the question, the original price increases by $ k $ percentage. Hence we will find the increase price which can expressed as:
\[\begin{array}{l}
{\rm{Increased \; price}} = 100 + \left( {\dfrac{k}{{100}} \times 100} \right)\\
{\rm{Increased \; price}} = 100 + k
\end{array}\]
 Now we will find the increased earnings since it is given in the question that earning increases by $ m $ percentage. This can be expressed as:
\[\begin{array}{l}
{\rm{Increased \; earning}} = 100 + \left( {\dfrac{m}{{100}} \times 100} \right)\\
{\rm{Increased \; earning}} = 100 + m
\end{array}\]
We will find the ratio of new price to new earning, which can be expressed as:
\[{\rm{New \; ratio}} = \dfrac{{100 + k}}{{100 + m}}\]
Now we will find the difference between the original and new ratio as represented below:
\[\begin{array}{l}
{\rm{Difference}} = \dfrac{{100 + k}}{{100 + m}} - 1\\
{\rm{Difference}} = \dfrac{{\left( {100 + k} \right) - \left( {100 + m} \right)}}{{100 + m}}\\
{\rm{Difference}} = \dfrac{{k - m}}{{100 + m}}
\end{array}\]
Now, we will find the percentage change in the ratios with respect to the original ratio, this can be represented as:
\[{\rm{Percentage \; change \; in \; ratio}} = \dfrac{{{\rm{Difference \; in \; ratio}}}}{{{\rm{Original \; ratio}}}} \times 100\]
 In the above expression we will substitute \[\dfrac{{k - m}}{{100 + m}}\] for difference in ratio and 1 for the original ratio. We get,
\[\begin{array}{l}
{\rm{Percentage \; change \; in \; ratio}} = \dfrac{{\dfrac{{k - m}}{{100 + m}}}}{1} \times 100\\
{\rm{Percentage \; change \; in \; ratio}} = \dfrac{{100\left( {k - m} \right)}}{{100 + m}}\%
\end{array}\]
So, the correct answer is “Option D”.

Note: In this question, we are using the concepts of ratios and percentages. This question is also based on the assumption as it will make it easy to proceed with the solution. When no data is given in the problem, and the percentage is given, we can always assume the original price as 100.