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Calculate Average Annual Compound Interest Rate for a $10,000 Loan Paid Back as $7,000 in 3 Years
Mathematics
Grade 11 (Senior High School)
Question Content
Page borrowed $10,000 from her aunt to donate to charity. At the end of three years, she paid back $7,000. What was the average annual compound rate of interest on Page's loan from her aunt?
Correct Answer
-11.21
Detailed Solution Steps
1
Step 1: Recall the compound interest formula for future value: $A = P(1 + r)^t$, where $A$ is the final amount paid back, $P$ is the principal (initial borrowed amount), $r$ is the annual compound interest rate, and $t$ is the time in years.
2
Step 2: Plug in the known values: $A = 7000$, $P = 10000$, $t = 3$. The equation becomes $7000 = 10000(1 + r)^3$.
3
Step 3: Isolate the exponential term by dividing both sides by 10000: $\\frac{7000}{10000} = (1 + r)^3$, which simplifies to $0.7 = (1 + r)^3$.
4
Step 4: Solve for $(1 + r)$ by taking the cube root of both sides: $1 + r = \\sqrt[3]{0.7}$. Calculate the cube root of 0.7, which is approximately 0.8879.
5
Step 5: Solve for $r$ by subtracting 1 from both sides: $r = 0.8879 - 1 = -0.1121$. Convert this to a percentage to get $r = -11.21\\%$.
Knowledge Points Involved
1
Compound Interest Formula
The formula $A = P(1 + r)^t$ calculates the future value $A$ of a principal amount $P$ invested or borrowed at an annual compound interest rate $r$ for $t$ years. It is used for scenarios where interest is calculated on both the initial principal and accumulated interest from previous periods. A negative rate indicates a decrease in the owed amount over time, as seen in this problem.
2
Root Operations for Solving Exponential Equations
When solving equations with variables in the exponent (like $(1 + r)^3 = 0.7$), we use root operations to isolate the variable. For an exponent of $n$, we take the $n$-th root of both sides. This is the inverse operation of raising a number to the power of $n$.
3
Negative Interest Rates
A negative interest rate means the principal amount owed decreases over time instead of increasing. In this context, it reflects that Page paid back less than she borrowed, effectively receiving a 'negative' interest (or a discount) on the loan.
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