- Translate the problem requirements: We need to find how the ratio (price per share ÷ earnings per share) changes when both the numerator and denominator increase by different percentages. This is asking for the percent change in a ratio when both parts of the ratio change.
- Set up concrete starting values: Use simple initial values for price and earnings per share to make calculations manageable and avoid working with complex algebraic expressions immediately.
- Calculate the new ratio after percentage increases: Apply the k% increase to price and m% increase to earnings, then find the new ratio value.
- Determine the percentage change in the ratio: Compare the original ratio to the new ratio using the standard percentage change formula, then generalize the result in terms of k and m.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what this question is really asking. We have a stock with two key metrics:
- Price per share (let's call this P)
- Earnings per share (let's call this E)
The question tells us that:
- Price increased by k percent
- Earnings increased by m percent
- k is greater than m
We need to find how much the ratio \(\mathrm{P/E}\) changed. This ratio is important in finance - it's called the Price-to-Earnings ratio or \(\mathrm{P/E}\) ratio.
Process Skill: TRANSLATE - Converting the financial scenario into a clear mathematical comparison problem
2. Set up concrete starting values
Instead of working with abstract variables right away, let's use simple numbers to make this easier to follow. Let's say:
- Original price per share = $100
- Original earnings per share = $10
This gives us an original \(\mathrm{P/E}\) ratio of \(\frac{100}{10} = 10\).
Using concrete numbers helps us see the pattern clearly before we generalize to the variables k and m.
3. Calculate the new ratio after percentage increases
Now let's apply the percentage increases:
New price per share:
- Increased by k percent means: New price = \(100 \times (1 + \frac{k}{100}) = 100 + k\)
New earnings per share:
- Increased by m percent means: New earnings = \(10 \times (1 + \frac{m}{100}) = (10 + \frac{10m}{100}) = (10 + \frac{m}{10})\)
New P/E ratio:
- New ratio = \((100 + k) \div (10 + \frac{m}{10})\)
- To make this cleaner: New ratio = \((100 + k) \div (10\frac{(100 + m)}{100}) = (100 + k) \times \frac{100}{10(100 + m)}\)
- Simplifying: New ratio = \(\frac{10(100 + k)}{(100 + m)}\)
4. Determine the percentage change in the ratio
Now we compare the original ratio to the new ratio:
- Original ratio = 10
- New ratio = \(\frac{10(100 + k)}{(100 + m)}\)
The percentage change formula is: \(\frac{(\text{New Value} - \text{Original Value})}{\text{Original Value}} \times 100\%\)
Percent change = \(\left[\frac{\frac{10(100 + k)}{(100 + m)} - 10}{10}\right] \times 100\%\)
Let's simplify step by step:
- = \(\left[\frac{\frac{10(100 + k)}{(100 + m)} - 10}{10}\right] \times 100\%\)
- = \(\left[\frac{10(100 + k) - 10(100 + m)}{10(100 + m)}\right] \times 100\%\)
- = \(\left[\frac{10(100 + k) - 10(100 + m)}{10(100 + m)}\right] \times 100\%\)
- = \(\left[\frac{10((100 + k) - (100 + m))}{10(100 + m)}\right] \times 100\%\)
- = \(\left[\frac{(100 + k - 100 - m)}{(100 + m)}\right] \times 100\%\)
- = \(\left[\frac{(k - m)}{(100 + m)}\right] \times 100\%\)
- = \(\frac{100(k - m)}{(100 + m)}\%\)
This result is independent of our choice of starting values - it works for any initial price and earnings values.
Process Skill: MANIPULATE - Careful algebraic simplification to reach the general formula
4. Final Answer
The ratio of price per share to earnings per share increased by \(\frac{100(k - m)}{(100 + m)}\%\).
Looking at our answer choices, this matches choice D. \(\frac{100(k - m)}{(100 + m)}\%\)
This makes intuitive sense: when the numerator of a ratio increases more than the denominator, the ratio increases. The exact amount depends on both the difference (k - m) and how much the denominator grew (100 + m).
Common Faltering Points
Errors while devising the approach
- Misunderstanding what ratio change means: Students often think they need to find the individual percentage changes and simply subtract them (k - m), missing that they need to find the percentage change of the ratio itself. The ratio P/E is a compound measure, and its percentage change requires comparing the new ratio to the original ratio.
- Confusion about percentage increase representation: Students may incorrectly represent a k% increase as just adding k instead of multiplying by \((1 + \frac{k}{100})\) or equivalently \(\frac{(100 + k)}{100}\). This leads to wrong expressions for the new price and earnings values.
- Not recognizing the need for concrete values: Students often jump straight into abstract algebraic manipulation without setting up clear initial values, making it much harder to track the ratio changes and leading to algebraic errors.
Errors while executing the approach
- Algebraic manipulation errors in percentage change formula: When applying the percentage change formula \(\frac{(\text{New} - \text{Original})}{\text{Original}} \times 100\%\), students frequently make errors in the algebraic simplification, particularly when dealing with the fraction \(\frac{\frac{10(100 + k)}{(100 + m)} - 10}{10}\). They may incorrectly combine terms or lose track of factors.
- Incorrect handling of denominators: Students often struggle with the step where they need to find a common denominator, especially when converting 10 to \(\frac{10(100 + m)}{(100 + m)}\) to subtract from \(\frac{10(100 + k)}{(100 + m)}\). This leads to computational errors.
- Sign errors in subtraction: When expanding \(10(100 + k) - 10(100 + m)\), students may incorrectly handle the distribution and subtraction, getting confused about which terms cancel out, potentially missing the negative sign in front of m.
Errors while selecting the answer
- Choosing the intuitive but incorrect (k - m)% option: After going through complex calculations, students may second-guess their work and select choice B, thinking "if price goes up more than earnings, the difference should just be k - m percent." This oversimplified reasoning ignores the compound nature of ratio changes.
- Confusing which variable goes in the denominator: Students may arrive at the correct form \(\frac{100(k - m)}{(100 + \text{something})}\) but incorrectly choose between (100 + k) and (100 + m) in the denominator, not recognizing that the denominator should reflect the change in the denominator of the original ratio (earnings), leading them to select choice C instead of D.
Alternate Solutions
Smart Numbers Approach
This problem involves percentage changes in ratios, which can be effectively solved using smart numbers by choosing convenient starting values.
Step 1: Choose smart starting values
Let's use simple, convenient numbers:
• Initial price per share = $100
• Initial earnings per share = $10
• Initial ratio = \(\frac{100}{10} = 10\)
We chose these values because:
• $100 makes percentage calculations straightforward
• $10 gives us a clean initial ratio of 10
• Both are easy to work with when applying percentage increases
Step 2: Apply specific percentage values
Let's use k = 20% and m = 10% (ensuring k > m as required)
• New price per share = $100 × 1.20 = $120
• New earnings per share = $10 × 1.10 = $11
• New ratio = \(\frac{120}{11}\)
Step 3: Calculate the percentage increase in ratio
• Original ratio = 10
• New ratio = \(\frac{120}{11} \approx 10.909\)
• Percentage increase = \(\left[\frac{\frac{120}{11} - 10}{10}\right] \times 100\%\)
• = \(\left[\frac{\frac{120}{11} - \frac{110}{11}}{10}\right] \times 100\%\)
• = \(\left[\frac{\frac{10}{11}}{10}\right] \times 100\%\)
• = \(\frac{1}{11} \times 100\% \approx 9.09\%\)
Step 4: Verify with the formula
Using answer choice D: \(\frac{100(k - m)}{(100 + m)}\%\)
• = \(\frac{100(20 - 10)}{(100 + 10)}\%\)
• = \(\frac{100(10)}{110}\%\)
• = \(\frac{1000}{110}\% \approx 9.09\%\)
The smart numbers approach confirms that answer choice D is correct!
