Question

If \$5,000 invested for one year at p percent simple annual interest yields \$500, what amount must be invested at k percent simple annual interest for one year to yield the same number of dollars?
(1) k = 0.8p
(2) k = 8

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

In some Data Sufficiency questions, the prompt itself contains enough information to solve for one or more variables. Always process this information first. Here, calculating \(p=10\) from the prompt is the key first step before looking at the statements.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves the simple interest formula. First, we need to use the given information to find the value of the interest rate \(p\). Then, we need to determine the principal amount for a second investment.
Step 2: Key Formula or Approach:
The formula for simple interest is \(I = P \cdot r \cdot t\), where \(I\) is the interest, \(P\) is the principal, \(r\) is the annual interest rate as a decimal, and \(t\) is the time in years.
Step 3: Detailed Explanation:
Part 1: Find the value of p.
For the first investment, we are given: \(P_1 = \$5,000\), \(I_1 = \$500\), \(t_1 = 1\) year, and the rate is \(p\) percent. \[ I_1 = P_1 \cdot \frac{p}{100} \cdot t_1 \] \[ 500 = 5000 \cdot \frac{p}{100} \cdot 1 \] \[ 500 = 50p \] \[ p = 10 \] So, the first interest rate was 10%.
Part 2: Answer the main question.
The question asks for the amount (\(P_2\)) that must be invested at \(k\) percent for one year to yield the same interest (\(I_2 = \$500\)). \[ I_2 = P_2 \cdot \frac{k}{100} \cdot t_2 \] \[ 500 = P_2 \cdot \frac{k}{100} \cdot 1 \] To find \(P_2\), we need the value of \(k\). So the question is essentially "What is the value of k?".
Analyze Statement (1): \(k = 0.8p\).
Since we found that \(p = 10\), we can calculate \(k\): \[ k = 0.8 \times 10 = 8 \] Now that we know \(k=8\), we can find \(P_2\): \[ 500 = P_2 \cdot \frac{8}{100} \implies P_2 = \frac{500 \cdot 100}{8} = \$6,250 \] Since we found a unique value for the amount, statement (1) is sufficient.
Analyze Statement (2): \(k = 8\).
This statement directly gives us the value of \(k\). We can find \(P_2\): \[ 500 = P_2 \cdot \frac{8}{100} \implies P_2 = \frac{500 \cdot 100}{8} = \$6,250 \] Since we found a unique value for the amount, statement (2) is sufficient.
Step 4: Final Answer:
Each statement alone provides enough information to determine the value of \(k\) and thus solve for the required investment amount.