- Translate the problem requirements: We need to find the total investment value after 18 years. The problem gives us a helpful rule: money doubles every \(\frac{70}{r}\) years when invested at r% interest. With 8% interest, money doubles every \(\frac{70}{8} = 8.75\) years.
- Determine the number of doubling periods: Calculate how many times the investment will double in 18 years by dividing the total time by the doubling period.
- Apply successive doubling: Starting with $5,000, apply the doubling effect for each complete period and handle any remaining partial period.
- Match with answer choices: Compare the calculated result with the given options to confirm the correct answer.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know and what we need to find:
- Starting investment: $5,000
- Interest rate: 8% per year, compounded annually
- Time period: 18 years
- We need to find: Total amount after 18 years
The problem gives us a very helpful rule: money doubles every \(\frac{70}{r}\) years when invested at r% interest. This is much easier than using complex compound interest formulas!
With 8% interest, the doubling time is: \(\frac{70}{8} = 8.75\) years
This means every 8.75 years, whatever amount we have will double.
Process Skill: TRANSLATE - Converting the problem's doubling rule into a concrete timeframe we can work with
2. Determine the number of doubling periods
Now we need to figure out how many times our money will double in 18 years.
We divide the total time by the doubling period:
Number of doubling periods = \(18 \div 8.75 = 2.06\) periods
This tells us the money will double completely twice, with a little bit of extra time left over (0.06 of a doubling period).
Since we're looking for an approximate answer and 0.06 is very small, we can consider this as exactly 2 complete doubling periods.
Process Skill: SIMPLIFY - Recognizing that 0.06 of a period is negligible for an approximation
3. Apply successive doubling
Let's trace what happens to our $5,000 investment:
- Start: $5,000
- After 1st doubling (≈8.75 years): $5,000 × 2 = $10,000
- After 2nd doubling (≈17.5 years): $10,000 × 2 = $20,000
Since 18 years is very close to 17.5 years (just 0.5 years difference), and we're looking for an approximate answer, our investment will be approximately $20,000.
4. Match with answer choices
Looking at our calculated result of $20,000, we can see this exactly matches choice (A) $20,000.
Let's do a quick reasonableness check: We started with $5,000 and it doubled twice, giving us 5,000 × 2 × 2 = 5,000 × 4 = $20,000. This confirms our answer.
Final Answer: (A) $20,000
Common Faltering Points
Errors while devising the approach
- Misunderstanding the doubling rule: Students may think the rule "\(\frac{70}{r}\)" applies only to specific interest rates or may not realize this is an approximation formula. They might attempt to use the exact compound interest formula \(\mathrm{A} = \mathrm{P}(1 + \mathrm{r})^\mathrm{t}\) instead of leveraging the simpler doubling concept given in the problem.
- Confusion about what "r" represents: Students may mistakenly use r = 0.08 (decimal form) instead of r = 8 (percentage form) when applying the \(\frac{70}{r}\) rule, leading to an incorrect doubling time of \(\frac{70}{0.08} = 875\) years instead of \(\frac{70}{8} = 8.75\) years.
Errors while executing the approach
- Rounding errors in doubling periods: When calculating \(18 \div 8.75 = 2.06\), students may incorrectly round this to 2.5 or 3 doubling periods instead of recognizing that 0.06 is negligible for an approximation problem, leading to incorrect final amounts.
- Arithmetic mistakes in successive doubling: Students may make basic multiplication errors when doubling: either calculating $5,000 × 2² incorrectly or losing track during the step-by-step doubling process ($5,000 → $10,000 → $20,000).
Errors while selecting the answer
- Not recognizing the approximate nature: Students who calculate the investment value at exactly 17.5 years (2 complete doubling periods) might think their answer of $20,000 is "too clean" and second-guess themselves, choosing a nearby option like $15,000 thinking they need to account for the extra 0.5 years more precisely.
