At the end of each year, the value of a certain antique watch is c percent more than its value...

1. Translate the problem requirements: The watch grows by the same percentage 'c' each year. We know its value on Jan 1, 1992 (k dollars) and Jan 1, 1994 (m dollars), and need to find its value on Jan 1, 1995.
2. Establish the growth pattern using the known values: Use the two-year period from 1992 to 1994 to understand how the value changes with the constant growth rate.
3. Identify the mathematical relationship: Recognize that if the value goes from k to m in 2 years with constant percentage growth, this creates a geometric progression.
4. Apply the pattern to find the 1995 value: Use the established relationship to determine what happens in the third year (1994 to 1995).

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

• The watch grows by the same percentage 'c' each year
• January 1, 1992: watch is worth k dollars
• January 1, 1994: watch is worth m dollars
• We need to find: value on January 1, 1995

The key insight is that the watch grows at a constant rate each year. Think of it like a savings account with the same interest rate applied every year.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Establish the growth pattern using the known values

Let's think about what happens year by year:

• Start of 1992: k dollars
• End of 1992 (start of 1993): \(\mathrm{k} \times \text{(some growth factor)}\)
• End of 1993 (start of 1994): m dollars

Since the percentage growth is the same each year, if we call the growth factor 'r' (where \(\mathrm{r} = 1 + \mathrm{c}/100\)), then:

After 1 year: \(\mathrm{k} \times \mathrm{r}\)
After 2 years: \(\mathrm{k} \times \mathrm{r} \times \mathrm{r} = \mathrm{k} \times \mathrm{r}^2\)

But we know that after 2 years, the value is m dollars. So:
\(\mathrm{k} \times \mathrm{r}^2 = \mathrm{m}\)

This tells us that \(\mathrm{r}^2 = \frac{\mathrm{m}}{\mathrm{k}}\), which means \(\mathrm{r} = \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\)

3. Identify the mathematical relationship

Now we can see the pattern clearly. Each year, the watch's value gets multiplied by the same factor \(\mathrm{r} = \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\).

Let's verify this makes sense:
• Start: k dollars
• After 1 year: \(\mathrm{k} \times \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\)
• After 2 years: \(\mathrm{k} \times \sqrt{\frac{\mathrm{m}}{\mathrm{k}}} \times \sqrt{\frac{\mathrm{m}}{\mathrm{k}}} = \mathrm{k} \times \frac{\mathrm{m}}{\mathrm{k}} = \mathrm{m}\) ✓

Perfect! This confirms our growth factor is correct.

Process Skill: INFER - Recognizing the geometric progression pattern from the given information

4. Apply the pattern to find the 1995 value

Now we can find the 1995 value by applying our growth factor one more time:

• January 1, 1994: m dollars
• January 1, 1995: \(\mathrm{m} \times \mathrm{r} = \mathrm{m} \times \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\)

Let's simplify this expression:
\(\mathrm{m} \times \sqrt{\frac{\mathrm{m}}{\mathrm{k}}} = \mathrm{m} \times \frac{\sqrt{\mathrm{m}}}{\sqrt{\mathrm{k}}} = \frac{\mathrm{m}\sqrt{\mathrm{m}}}{\sqrt{\mathrm{k}}}\)

This matches answer choice C exactly!

Let's double-check by working from the beginning:
• 1992: k
• 1993: \(\mathrm{k} \times \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\)
• 1994: \(\mathrm{k} \times \left(\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\right)^2 = \mathrm{k} \times \frac{\mathrm{m}}{\mathrm{k}} = \mathrm{m}\) ✓
• 1995: \(\mathrm{m} \times \sqrt{\frac{\mathrm{m}}{\mathrm{k}}} = \frac{\mathrm{m}\sqrt{\mathrm{m}}}{\sqrt{\mathrm{k}}}\) ✓

4. Final Answer

The value of the watch on January 1, 1995 is \(\frac{\mathrm{m}\sqrt{\mathrm{m}}}{\sqrt{\mathrm{k}}}\) dollars.

This corresponds to answer choice C.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the time intervals
Students often confuse "at the end of each year" with the January 1st dates given. They might think the growth happens on January 1st rather than understanding that January 1, 1994 represents the value after 2 full years of growth from January 1, 1992. This leads to setting up equations with incorrect time periods.

2. Assuming linear growth instead of compound growth
Many students interpret "c percent more each year" as simple interest rather than compound interest. They might think the value increases by the same dollar amount each year (\(\mathrm{k} + \mathrm{cx}, \mathrm{k} + 2\mathrm{cx}\), etc.) instead of recognizing that each year's growth is applied to the previous year's total value, creating a geometric progression.

3. Confusing which values correspond to which years
Students may mix up whether k and m represent beginning-of-year or end-of-year values, or they might incorrectly associate these values with the wrong years, leading to an incorrect setup of the growth equation.

Errors while executing the approach

1. Algebraic manipulation errors when solving for the growth factor
When establishing that \(\mathrm{k} \times \mathrm{r}^2 = \mathrm{m}\), students often make mistakes isolating r. Common errors include forgetting to take the square root (leaving \(\mathrm{r}^2 = \frac{\mathrm{m}}{\mathrm{k}}\)) or incorrectly applying the square root to get \(\mathrm{r} = \frac{\sqrt{\mathrm{m}}}{\sqrt{\mathrm{k}}}\) instead of \(\mathrm{r} = \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\).

2. Incorrect simplification of radical expressions
When calculating the final expression \(\mathrm{m} \times \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\), students frequently struggle with the radical manipulation. They might incorrectly write this as \(\frac{\mathrm{m}\sqrt{\mathrm{m}}}{\mathrm{k}}\) instead of \(\frac{\mathrm{m}\sqrt{\mathrm{m}}}{\sqrt{\mathrm{k}}}\), or fail to recognize that \(\mathrm{m} \times \sqrt{\frac{\mathrm{m}}{\mathrm{k}}} = \frac{\mathrm{m}\sqrt{\mathrm{m}}}{\sqrt{\mathrm{k}}}\) through proper radical rules.

3. Sign and calculation errors in verification steps
Students may make arithmetic errors when checking their work by substituting back into the original relationships, leading them to doubt a correct answer or accept an incorrect one.

Errors while selecting the answer

1. Mismatching equivalent forms of the same expression
Students may arrive at the correct mathematical expression but fail to recognize it matches answer choice C due to different formatting. For example, they might write their answer as \(\mathrm{m} \cdot \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}\) and not realize this equals \(\frac{\mathrm{m}\sqrt{\mathrm{m}}}{\sqrt{\mathrm{k}}}\), leading them to think none of the choices are correct.

2. Choosing a superficially similar but incorrect answer
Answer choices A and B contain familiar elements like \((\mathrm{m}-\mathrm{k})\) terms that might appeal to students who tried linear approaches. Students who realize their linear approach was wrong might still gravitate toward these choices because they seem more intuitive than the radical expression that is actually correct.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose smart numbers for k and the growth rate

Let's set \(\mathrm{k} = 100\) dollars (the value on January 1, 1992). This makes percentage calculations clean.

For the growth rate, let's choose \(\mathrm{c} = 44\%\), so the multiplier each year is \(1.44\).

Step 2: Calculate m using our chosen values

From 1992 to 1994 (2 years of growth):

Value on Jan 1, 1994: \(\mathrm{m} = \mathrm{k} \times (1.44)^2 = 100 \times 2.0736 = 207.36\)

So we have: \(\mathrm{k} = 100, \mathrm{m} = 207.36\)

Step 3: Calculate the value on January 1, 1995

Value on Jan 1, 1995 = \(\mathrm{m} \times 1.44 = 207.36 \times 1.44 = 298.5984\)

Step 4: Test each answer choice with our smart numbers

• Choice A: \(\mathrm{m} + \frac{1}{2}(\mathrm{m}-\mathrm{k}) = 207.36 + \frac{1}{2}(107.36) = 207.36 + 53.68 = 261.04 \neq 298.5984\)
• Choice B: \(\mathrm{m} + \frac{1}{2} \times \frac{\mathrm{m}-\mathrm{k}}{\mathrm{k}} \times \mathrm{m} = 207.36 + \frac{1}{2} \times \frac{107.36}{100} \times 207.36 = 207.36 + 111.25 = 318.61 \neq 298.5984\)
• Choice C: \(\frac{\mathrm{m}\sqrt{\mathrm{m}}}{\sqrt{\mathrm{k}}} = \frac{207.36 \times \sqrt{207.36}}{\sqrt{100}} = \frac{207.36 \times 14.4}{10} = \frac{2985.984}{10} = 298.5984\) ✓
• Choice D: \(\frac{\mathrm{m}^2}{2\mathrm{k}} = \frac{(207.36)^2}{200} = \frac{42998.17}{200} = 214.99 \neq 298.5984\)
• Choice E: \(\mathrm{k}\mathrm{m}^2 = 100 \times (207.36)^2 = 4,299,817 \neq 298.5984\)

Step 5: Verify the pattern

Notice that \((1.44)^2 = 2.0736\), so \(\sqrt{2.0736} = 1.44\), which confirms our growth multiplier.

The smart numbers approach clearly shows that Choice C is correct.