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Kylie invested a certain amount of money at \(\mathrm{r}\%\) yearly interest compounded at the end of each year and the same amount of money at \((\mathrm{r} + 2)\%\) yearly interest compounded at the end of each year. What was the amount of money that Kylie invested at the \((\mathrm{r} + 2)\%\) yearly interest rate?
Let's break down what we're being asked. Kylie made two investments of equal amounts – one at r% interest and another at (r+2)% interest. Both compound annually. We need to find the specific dollar amount she invested at the (r+2)% rate.
Since Kylie invested the same amount at both rates, finding this principal amount (let's call it P) directly answers our question.
For this value question to be sufficient, we need enough information to determine the exact value of P.
Statement 1 tells us that after 1 year, the investment at (r+2)% earned $200 more in interest than the investment at r%.
Let's think about what this means. The interest earned at:
The difference in interest is:
\(\mathrm{P} \times \frac{(\mathrm{r}+2)}{100} - \mathrm{P} \times \frac{\mathrm{r}}{100} = \$200\)
Simplifying this:
\(\mathrm{P} \times \frac{[(\mathrm{r}+2) - \mathrm{r}]}{100} = \$200\)
\(\mathrm{P} \times \frac{2}{100} = \$200\)
\(\mathrm{P} \times 0.02 = \$200\)
\(\mathrm{P} = \$10,000\)
[STOP - Sufficient!] Statement 1 gives us a unique value: P = $10,000.
Statement 1 is sufficient.
This eliminates choices B, C, and E.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us that r = 3.
This means the two interest rates are 3% and 5%. However, knowing just the interest rates tells us nothing about how much money Kylie invested. She could have invested $1,000, $10,000, or $100,000 – we have no way to determine the specific amount.
For example:
Without knowing the actual interest difference, we cannot determine P.
Statement 2 is NOT sufficient.
This eliminates choices B and D.
Since Statement 1 alone is sufficient but Statement 2 alone is not, the answer is A.
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."