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A bank account earned \(2\%\) annual interest, compounded daily, for as long as the balance was under \(\$1{,}000\), starting when the account was opened. Once the balance reached \(\$1{,}000\), the account earned \(2.5\%\) annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the \(2\%\) rate greater than the total amount earned at the \(2.5\%\) rate?
Let's break down what we're being asked. We have a bank account that:
The question asks: Was the total interest earned at \(2\%\) rate > total interest earned at \(2.5\%\) rate?
This is a yes/no question. We need to find whether we can definitively answer "yes" or "no."
Here's the crucial relationship to understand:
Think about it: The longer the account takes to reach \(\$1,000\), the smaller the initial deposit must have been. And if the initial deposit was smaller, more interest was earned at the \(2\%\) rate.
Statement 1: The account earned exactly \(\$25\) in interest at the \(2.5\%\) rate.
This tells us that the final balance = \(\$1,025\) (since the balance was \(\$1,000\) when the rate switched).
We don't know:
Let's consider what could happen with different time periods:
Scenario 1: Account spent many years at \(2\%\) rate, short time at \(2.5\%\) rate
Scenario 2: Account spent short time at \(2\%\) rate, longer time at \(2.5\%\) rate
Since we get different answers (YES vs NO), Statement 1 is NOT sufficient.
This eliminates choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: The account was open for exactly three years.
This tells us that the total time at both rates equals 3 years. But how those 3 years were split makes all the difference!
We don't know:
Scenario 1: 2.9 years at \(2\%\), 0.1 years at \(2.5\%\)
Scenario 2: 0.1 years at \(2\%\), 2.9 years at \(2.5\%\)
These scenarios give opposite answers, so Statement 2 is NOT sufficient.
This eliminates choice B.
Now let's see what happens when we use both statements together.
From Statement 1: Interest at \(2.5\%\) = \(\$25\) (so final balance = \(\$1,025\))
From Statement 2: Total time = 3 years
Here's the key insight: With both pieces of information, there's only ONE possible time split that would result in exactly \(\$25\) interest at the \(2.5\%\) rate.
Think about it this way:
This means:
We can now definitively answer YES to the question.
[STOP - Sufficient!] Both statements together give us a definitive answer.
This eliminates choice E.
Both statements together are sufficient to answer the question, but neither statement alone is sufficient.
Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."