Kate invested x dollars, where x geq 1{,}000, for one year in a new account that earned interest at an...
GMAT Data Sufficiency : (DS) Questions
Kate invested \(\mathrm{x}\) dollars, where \(\mathrm{x} \geq 1{,}000\), for one year in a new account that earned interest at an annual rate of \(\mathrm{r}\) percent, compounded semiannually. Bob invested \(\mathrm{y}\) dollars for one year in a new account that earned interest at an annual rate of 5 percent, compounded annually. If there were no other transactions to these accounts and if the sum consisting of the amount invested and the interest earned for the year was the same for both accounts, what was the value of \(\mathrm{r}\)?
- \(\mathrm{y} = 1.05\mathrm{x}\)
- \(\mathrm{y} = \mathrm{x} + 100\)
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."