Multiple Choice
If you take out a loan of \$678 at an annual interest rate of 10.5% for one year, how much interest will you pay at the end of the year (assuming simple interest)?
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
To what amount will $P invested for n years at an annual interest rate of r percent, compounded annually, accumulate?
A
\(P(1 + r)^n\)
B
\(P(1 + n)^r\)
C
\(P \times r \times n\)
D
\(P(1 + r \, n)\)
Verified step by step guidance1
Step 1: Understand the concept of compound interest. Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. The formula for compound interest is: \( A = P(1 + r)^n \), where \( A \) is the accumulated amount, \( P \) is the principal, \( r \) is the annual interest rate (in decimal form), and \( n \) is the number of years.
Step 2: Analyze the given options. The correct formula for compound interest is \( P(1 + r)^n \). The other options provided do not correctly represent the compounding process. For example, \( P(1 + n)^r \) incorrectly applies the interest rate as an exponent to the time period, and \( P \times r \times n \) represents simple interest, not compound interest.
Step 3: Break down the correct formula \( P(1 + r)^n \). Here, \( (1 + r) \) represents the growth factor for one year, and raising it to the power of \( n \) accounts for compounding over \( n \) years.
Step 4: Convert the annual interest rate \( r \) from a percentage to a decimal by dividing it by 100. For example, if \( r \) is 5%, then \( r = 0.05 \). This step ensures the formula is applied correctly.
Step 5: Substitute the values of \( P \), \( r \), and \( n \) into the formula \( A = P(1 + r)^n \) to calculate the accumulated amount. Ensure all values are correctly placed and the exponentiation is performed accurately.
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