Question

Pat invested $4,000 at a percent simple annual interest and a different amount at p percent simple annual interest for the same period of time. What amount did Pat invest at p percent simple annual interest?
(1) The total amount of interest earned by Pat's investments in one year was $400.
(2) Pat invested the $4,000 at 4 percent simple annual interest.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

For data sufficiency problems involving multiple variables, count your independent equations and your variables. To find a unique solution for a variable, you generally need as many independent equations as you have variables. Here, even after combining, we have two variables (\(P_2\), \(p\)) but only one equation.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
We are dealing with simple interest from two investments. The formula for simple interest is:
\[ I = P \cdot r \cdot t \] where \(P\) is the principal, \(r\) is the annual rate, and \(t\) is the time in years.

We need to determine the value of \(P_2\).

Step 2: Defining the Variables:
- Investment 1: Principal \(P_1 = 4000\), rate = \(a\%\).
- Investment 2: Principal \(P_2\), rate = \(p\%\).
- Time = 1 year.
- Total interest = 400.

Step 3: Analysis of Statement (1)
Total interest in 1 year = 400.
\[ 400 = (4000 \cdot \tfrac{a}{100}) + (P_2 \cdot \tfrac{p}{100}) \] \[ 400 = 40a + \frac{P_2 \cdot p}{100} \] This is one equation with three unknowns (\(a, P_2, p\)).
We cannot uniquely determine \(P_2\).
Statement (1) alone is not sufficient.

Step 4: Analysis of Statement (2)
This statement tells us \(a = 4\%\).
But it provides no information about \(P_2, p,\) or total interest.
Statement (2) alone is not sufficient.

Step 5: Combining Statements (1) and (2)
From (1): \[ 400 = 40a + \frac{P_2 \cdot p}{100} \] Substitute \(a = 4\): \[ 400 = 40(4) + \frac{P_2 \cdot p}{100} \] \[ 400 = 160 + \frac{P_2 \cdot p}{100} \] \[ 240 = \frac{P_2 \cdot p}{100} \] \[ P_2 \cdot p = 24000 \] Even with both statements, we only have a product of \(P_2\) and \(p\), not the individual value of \(P_2\).
For example:
- If \(p=8\%\), then \(P_2 = 3000\).
- If \(p=6\%\), then \(P_2 = 4000\).
Thus, multiple values are possible.
Statements (1) and (2) together are not sufficient.

Step 6: Final Answer
Neither statement alone nor both together are sufficient to determine \(P_2\).
Correct Option: (E)