Kate invested x dollars, where x geq 1{,}000, for one year in a new account that earned interest at an...

Understanding the Question

We need to find the value of r, Kate's annual interest rate.

Given Information

• Kate invested x dollars (\(\mathrm{x} \geq 1,000\)) at r% annual rate, compounded semiannually
• Bob invested y dollars at 5% annual rate, compounded annually
• Both investments were for one year
• After one year, both accounts had the same total value (principal + interest)

Setting Up the Problem

Since Kate's money compounds semiannually at r% annual rate, she earns \(\frac{\mathrm{r}}{200}\) per 6-month period. After two periods, Kate's final amount is \(\mathrm{x}(1 + \frac{\mathrm{r}}{200})^2\). Bob's final amount is simply \(\mathrm{y}(1.05)\).

These must be equal: \(\mathrm{x}(1 + \frac{\mathrm{r}}{200})^2 = \mathrm{y}(1.05)\)

We have three unknowns (x, y, r) but only one equation. To find a unique value of r, we need additional information that creates enough constraints.

Analyzing Statement 1

Statement 1: \(\mathrm{y} = 1.05\mathrm{x}\) (Bob invested exactly 5% more than Kate)

This creates an elegant pattern! Let's think about what happens:

• Bob starts with: \(1.05\mathrm{x}\) (5% more than Kate)
• Bob's money grows by: \(1.05\) (his 5% interest rate)
• Bob ends with: \((1.05\mathrm{x})(1.05) = 1.05^2\mathrm{x} = 1.1025\mathrm{x}\)

For Kate to end with the same amount starting from just x, her growth factor must be 1.1025.

Since Kate's growth factor equals \((1 + \frac{\mathrm{r}}{200})^2\):

• \((1 + \frac{\mathrm{r}}{200})^2 = 1.1025\)
• Notice that \(1.1025 = 1.05^2\)
• Therefore: \(1 + \frac{\mathrm{r}}{200} = 1.05\)
• So: \(\frac{\mathrm{r}}{200} = 0.05\)
• Which gives us: \(\mathrm{r} = 10\)

We get exactly one value for r.

[STOP - Statement 1 is SUFFICIENT!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: \(\mathrm{y} = \mathrm{x} + 100\) (Bob invested $100 more than Kate)

The key insight: this $100 represents different percentage advantages depending on the investment size.

Testing Different Investment Amounts

When x = $1,000:

• Bob invests y = $1,100
• Bob's advantage: \(\frac{\$100}{\$1,000} = 10\%\)
• Bob ends with: \(\$1,100 \times 1.05 = \$1,155\)
• Kate needs: \(\mathrm{x}(1 + \frac{\mathrm{r}}{200})^2 = \$1,155\)
• Kate's growth factor: \(\frac{\$1,155}{\$1,000} = 1.155\)

When x = $2,000:

• Bob invests y = $2,100
• Bob's advantage: \(\frac{\$100}{\$2,000} = 5\%\)
• Bob ends with: \(\$2,100 \times 1.05 = \$2,205\)
• Kate needs: \(\mathrm{x}(1 + \frac{\mathrm{r}}{200})^2 = \$2,205\)
• Kate's growth factor: \(\frac{\$2,205}{\$2,000} = 1.1025\)

Different investment amounts require different growth factors for Kate, which means different values of r.

Statement 2 is NOT SUFFICIENT.

This eliminates choices B and D.

The Answer: A

Statement 1 creates a beautiful mathematical relationship where Bob's percentage advantage (5%) exactly matches his interest rate, allowing us to determine \(\mathrm{r} = 10\) uniquely. Statement 2's fixed dollar difference leads to different required rates depending on the investment amount.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."