A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1{,}000, starting when...

Understanding the Question

Let's break down what we're being asked. We have a bank account that:

• Started with some initial deposit
• Earned \(2\%\) annual interest (compounded daily) while balance < \(\$1,000\)
• Once it reached \(\$1,000\), switched to \(2.5\%\) annual interest (compounded daily)
• Had no deposits or withdrawals
• Eventually closed

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."

Key Insight

Here's the crucial relationship to understand:

• Interest earned at \(2\%\) = \(\$1,000\) - Initial deposit
• Interest earned at \(2.5\%\) = Final balance - \(\$1,000\)

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.

Analyzing Statement 1

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).

What We Still Don't Know

We don't know:

• How long the account was at the \(2.5\%\) rate
• The initial deposit amount
• How long the account was at the \(2\%\) rate

Testing Different Scenarios

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

• If it took 5 years to grow from \(\$900\) to \(\$1,000\) at \(2\%\), then interest at \(2\%\) = \(\$100\)
• The remaining short time at \(2.5\%\) earned \(\$25\)
• Answer to question: YES (\(\$100 > \$25\))

Scenario 2: Account spent short time at \(2\%\) rate, longer time at \(2.5\%\) rate

• If it took only 6 months to grow from \(\$980\) to \(\$1,000\) at \(2\%\), then interest at \(2\%\) = \(\$20\)
• The remaining longer time at \(2.5\%\) earned \(\$25\)
• Answer to question: NO (\(\$20 < \$25\))

Since we get different answers (YES vs NO), Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

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!

What We Still Don't Know

We don't know:

• How the 3 years were split between the two rates
• The initial deposit
• The final balance

Testing Different Scenarios

Scenario 1: 2.9 years at \(2\%\), 0.1 years at \(2.5\%\)

• To take 2.9 years to reach \(\$1,000\) at \(2\%\), the initial deposit must be very small
• Interest at \(2\%\) would be large (close to \(\$1,000\) minus that small deposit)
• Interest at \(2.5\%\) would be minimal (only 0.1 years of growth)
• Answer to question: YES

Scenario 2: 0.1 years at \(2\%\), 2.9 years at \(2.5\%\)

• To reach \(\$1,000\) in just 0.1 years at \(2\%\), the initial deposit must be close to \(\$1,000\)
• Interest at \(2\%\) would be minimal (maybe just a few dollars)
• Interest at \(2.5\%\) would be substantial (2.9 years of growth from \(\$1,000\))
• Answer to question: NO

These scenarios give opposite answers, so Statement 2 is NOT sufficient.

This eliminates choice B.

Combining Both Statements

Now let's see what happens when we use both statements together.

Combined Information

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.

Why Both Together Are Sufficient

Think about it this way:

• \(\$25\) is relatively modest interest for the \(2.5\%\) rate
• If the account had spent most of its 3 years at \(2.5\%\) (say 2.5 years), it would have earned much more than \(\$25\)
• To earn only \(\$25\) at \(2.5\%\), the account must have spent a relatively short time at this rate

This means:

• The account spent MOST of the 3 years at the \(2\%\) rate
• Therefore, the initial deposit was small
• Therefore, the interest earned at \(2\%\) was substantial (close to \(\$1,000\) minus a small initial deposit)
• This interest at \(2\%\) is definitely greater than \(\$25\)

We can now definitively answer YES to the question.

[STOP - Sufficient!] Both statements together give us a definitive answer.

This eliminates choice E.

The Answer: C

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."