The function \(\mathrm{f(x) = 206(1.034)^x}\) models the value, in dollars, of a certain bank account by the end of each...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{f(x) = 206(1.034)^x}\) models the value, in dollars, of a certain bank account by the end of each year from 1957 through 1972, where x is the number of years after 1957. Which of the following is the best interpretation of '\(\mathrm{f(5) ≈ 243}\)' in this context?
The value of the bank account is estimated to be approximately \(5\) dollars greater in \(1962\) than in \(1957\).
The value of the bank account is estimated to be approximately \(243\) dollars in \(1962\).
The value, in dollars, of the bank account is estimated to be approximately \(5\) times greater in \(1962\) than in \(1957\).
The value of the bank account is estimated to increase by approximately \(243\) dollars every \(5\) years between \(1957\) and \(1972\).
1. TRANSLATE the given information
- Given function: \(\mathrm{f(x) = 206(1.034)^x}\)
- Context: Models bank account value in dollars from 1957-1972
- Variable meaning: \(\mathrm{x}\) = number of years after 1957
- Statement to interpret: "\(\mathrm{f(5)}\) is approximately equal to 243"
2. TRANSLATE what \(\mathrm{f(5)}\) means
- \(\mathrm{f(5)}\) means we substitute \(\mathrm{x = 5}\) into the function
- \(\mathrm{x = 5}\) represents 5 years after 1957
- \(\mathrm{1957 + 5 = 1962}\)
- So \(\mathrm{f(5)}\) represents the bank account value in the year 1962
3. INFER the complete interpretation
- If \(\mathrm{f(5) ≈ 243}\), this means when \(\mathrm{x = 5}\) (year 1962), the function output is approximately 243
- Since the function outputs bank account values in dollars
- Therefore: The bank account value in 1962 is approximately $243
4. TRANSLATE to match answer choices
- Looking for: "value is approximately 243 dollars in 1962"
- Choice A: Talks about $5 difference - not what \(\mathrm{f(5) = 243}\) tells us
- Choice B: "Value is approximately 243 dollars in 1962" - matches our interpretation
- Choice C: Talks about 5 times greater - not what \(\mathrm{f(5) = 243}\) tells us
- Choice D: Talks about rate per 5 years - not what \(\mathrm{f(5) = 243}\) tells us
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skills: Students confuse what the numbers 5 and 243 represent in the context. They might think:
- The "5" in \(\mathrm{f(5)}\) refers to a $5 amount rather than 5 years after 1957
- The "243" refers to a rate of change or difference rather than the actual account value
- They don't connect \(\mathrm{f(5)}\) to the specific year 1962
This leads them to misinterpret the statement and may cause them to select Choice A (5 dollars greater) or Choice D (243 dollars every 5 years).
Second Most Common Error:
Poor INFER reasoning: Students understand that \(\mathrm{f(5)}\) means substituting \(\mathrm{x = 5}\), but fail to connect this to what it means in the real-world context. They might correctly identify that \(\mathrm{x = 5}\) means 5 years after 1957, but don't complete the reasoning to realize this corresponds to 1962.
This incomplete reasoning leads to confusion and guessing among the remaining choices.
The Bottom Line:
Function notation problems require careful translation between mathematical symbols and real-world meaning. Students must connect \(\mathrm{f(5)}\) to both the input value (5 years after 1957 = 1962) AND the output value (243 dollars), then match this complete interpretation to the answer choices.
The value of the bank account is estimated to be approximately \(5\) dollars greater in \(1962\) than in \(1957\).
The value of the bank account is estimated to be approximately \(243\) dollars in \(1962\).
The value, in dollars, of the bank account is estimated to be approximately \(5\) times greater in \(1962\) than in \(1957\).
The value of the bank account is estimated to increase by approximately \(243\) dollars every \(5\) years between \(1957\) and \(1972\).