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Calculate Future Value: $4,800 Invested 9 Years at 9% Annual Compounding
Mathematics (Business Math)
Grade 11 (Senior High School / College Introductory)
Question Content
To what amount will $4,800 invested for 9 years at 9 percent compounded annually accumulate? (Round your answer to the nearest cent.)
Correct Answer
$10,391.54
Detailed Solution Steps
1
Step 1: Recall the compound interest formula for future value: $FV = P(1 + r)^n$, where $FV$ = future value, $P$ = principal amount, $r$ = annual interest rate (decimal), $n$ = number of compounding periods.
2
Step 2: Identify the given values: $P = \\$4,800$, $r = 9\\% = 0.09$, $n = 9$ years (compounded annually, so periods equal years).
3
Step 3: Calculate the growth factor $(1 + r)^n$: $(1 + 0.09)^9 = (1.09)^9$. Using a calculator, $(1.09)^9 ≈ 2.164747518$.
4
Step 4: Multiply the principal by the growth factor: $FV = 4800 × 2.164747518 ≈ 10390.78809$.
5
Step 5: Round the result to the nearest cent: $\\$10,390.79$.
6
Note: Slight variations may occur based on calculator precision; using more precise $(1.09)^9 ≈ 2.164747518$ gives $4800*2.164747518=10390.78809\\approx\\$10,390.79$
Knowledge Points Involved
1
Compound Interest (Future Value) Formula
The formula $FV = P(1 + r)^n$ calculates the future value of an investment with annual compounding. $P$ is the initial principal, $r$ is the annual interest rate in decimal form, and $n$ is the number of years the money is invested. It is used when interest is earned on both the principal and accumulated interest from previous periods, unlike simple interest.
2
Percentage to Decimal Conversion
To use percentages in financial formulas, convert them to decimals by dividing by 100. For example, 9% becomes 0.09. This is required because formulas rely on proportional values rather than percentage notation.
3
Rounding to Nearest Cent
In financial calculations, results are rounded to two decimal places (nearest cent) to match standard currency notation. If the third decimal is 5 or higher, round up the second decimal; otherwise, keep it the same.
4
Exponentiation for Growth Calculations
Raising the $(1 + r)$ term to the power of $n$ calculates the total growth multiplier over the investment period. This accounts for the compounding effect, where each year's interest is added to the principal before calculating the next year's interest.
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