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75% (01:18) correct
25%
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If $P were invested at the annual interest rate of 16 percent compounded semiannually, what would be the total value of the investment at the end of x years?
(1) (1+ \(\frac{8}{100}\))^6x = 27
(2) P = 10,000
(1) (1+ \(\frac{8}{100}\))^6x = 27
(2) P = 10,000
17 Mar 2025, 12:43
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Step 1: Formula for Compound Interest
Since the interest is compounded semiannually, the formula for compound interest is:
\(\\
A = P \left(1 + \frac{r}{100}\right)^{n}\\
\\
\)
where:
• A is the final amount,
• P is the first amount,
• r = 8% (16% annual interest rate so 8% semi-annual),
• n = 2x (x year annual so 2x year compounded semiannually),
Thus, the formula simplifies to:
\(\\
A = P \left(1 + \frac{8}{100} \right)^{2x}\\
\\
\)
Step 2: Analyzing the Given Statements
Statement (1):
\(\\
\left(1 + \frac{8}{100} \right)^{6x}= 27\\
\)
\(\\
\left(1 + \frac{8}{100} \right)^{2x}= 3\\
\)
Since x is now known, we can substitute into A = 3P, but we still need P.
This alone is insufficient
Statement (2):
P = 10,000
This alone is insufficient to determine because we do not know x.
Step 3: Combining Both Statements
From (1), we determined x, and from (2), we have P. Substituting both into:
A = 3 * 10,000
Since we can calculate x from (1), the final amount A can now be uniquely determined.
Thus, both statements together are sufficient to determine A(final amount), but individually, they are not.
Final Answer:C (Both statements together are sufficient, but neither alone is sufficient.)
Since the interest is compounded semiannually, the formula for compound interest is:
\(\\
A = P \left(1 + \frac{r}{100}\right)^{n}\\
\\
\)
where:
• A is the final amount,
• P is the first amount,
• r = 8% (16% annual interest rate so 8% semi-annual),
• n = 2x (x year annual so 2x year compounded semiannually),
Thus, the formula simplifies to:
\(\\
A = P \left(1 + \frac{8}{100} \right)^{2x}\\
\\
\)
Step 2: Analyzing the Given Statements
Statement (1):
\(\\
\left(1 + \frac{8}{100} \right)^{6x}= 27\\
\)
\(\\
\left(1 + \frac{8}{100} \right)^{2x}= 3\\
\)
Since x is now known, we can substitute into A = 3P, but we still need P.
This alone is insufficient
Statement (2):
P = 10,000
This alone is insufficient to determine because we do not know x.
Step 3: Combining Both Statements
From (1), we determined x, and from (2), we have P. Substituting both into:
A = 3 * 10,000
Since we can calculate x from (1), the final amount A can now be uniquely determined.
Thus, both statements together are sufficient to determine A(final amount), but individually, they are not.
Final Answer:C (Both statements together are sufficient, but neither alone is sufficient.)
siddhantvarma
