Question

Helena invested 8000 in the Tallahassee City Bank at z% simple annual interest for one year with a yield of $450. How much should she invest at s% simple annual interest for one year to yield the same amount?
(1) \( \frac{s}{100} = \frac{3}{4} \)
(2) \( s = 0.4z \)

• 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:

In Data Sufficiency, you don't always need to compute the final numerical answer. You just need to be certain that a unique answer *can* be computed. Once you determine that a statement provides a unique value for a necessary variable (like 's' in this case), you can conclude it is sufficient.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This Data Sufficiency problem asks for a specific value (the amount of a second investment). To solve it, we need to determine if the given statements provide enough information to calculate this value.
Step 2: Key Formula or Approach:
The formula for simple interest is \(I = P \times r \times t\), where \(I\) is the interest, \(P\) is the principal, \(r\) is the annual interest rate, and \(t\) is the time in years.
Let \(P_s\) be the principal of the second investment. The question asks for the value of \(P_s\).
For the second investment, the yield (interest) is the same, \$450. The rate is s%, and the time is 1 year.
So, the equation is: \(450 = P_s \times \frac{s}{100} \times 1\).
To find \(P_s\), we need to determine the value of \(s\).
Step 3: Detailed Explanation:
First, let's use the information given in the main question to find the value of \(z\).
\(450 = 8000 \times \frac{z}{100} \times 1\)
\(450 = 80z\)
\(z = \frac{450}{80} = \frac{45}{8} = 5.625\)
Now let's analyze the statements to see if they determine the value of \(s\).
Analyzing Statement (1):
"s/100 = 3/4"
This equation directly gives us the value of the rate for the second investment.
\(s = 100 \times \frac{3}{4} = 75\).
Since we have a specific value for \(s\), we can plug it into our equation for the second investment and solve for \(P_s\):
\(450 = P_s \times \frac{75}{100}\).
\(P_s = 450 \times \frac{100}{75} = 600\).
Since we can find a unique value for \(P_s\), statement (1) is sufficient.
Analyzing Statement (2):
"s = .4z"
We already calculated the value of \(z\) from the initial information: \(z = 45/8\).
We can use this to find \(s\):
\(s = 0.4 \times \frac{45}{8} = \frac{4}{10} \times \frac{45}{8} = \frac{2}{5} \times \frac{45}{8} = \frac{90}{40} = \frac{9}{4} = 2.25\).
Since we have a specific value for \(s\), we can solve for \(P_s\):
\(450 = P_s \times \frac{2.25}{100}\).
We don't need to calculate \(P_s\), just know that it can be calculated uniquely.
Therefore, statement (2) is also sufficient.
Step 4: Final Answer:
Each statement alone provides enough information to determine the amount that should be invested.