ASVAB Study Guide4.Percent Change
Calculating the total accumulated balance where interest grows on both the principal and past interest.
$$A = P \times (1 + R)^T$$
Interest is calculated on the initial principal and also on the accumulated interest of all previous periods.
Example
If you invest $100 at a 10% interest rate compounded annually, what is the total balance after 2 years?
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Step 1: Convert 10% to a decimal: $$\frac{10}{100} = 0.10$$
Step 2: Add to 1 for growth factor: $$1 + 0.10 = 1.10$$
Step 3: Calculate value: $$100 \times (1.10)^2 = 100 \times 1.21 = 121$$
Practice problems
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Problem 1
A savings account holds $1000 and earns 5% interest compounded annually. What is the total balance in the account after 2 years?
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Answer: $1102.50
Step 1: Convert 5% to a decimal: 0.05. Growth multiplier = $$1 + 0.05 = 1.05$$
Step 2: Apply exponents for time periods: $$1000 \times (1.05)^2$$
Step 3: Calculate total: 1000 × 1.1025 = $1102.50
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Problem 2
You invest $500 at an annual rate of 6% compounded annually. How much interest (excluding principal) has accumulated after 2 years?
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Answer: $61.80
Step 1: Growth multiplier = $$1 + 0.06 = 1.06$$. Calculate total: $$500 \times (1.06)^2$$
Step 2: Find account value: 500 × 1.1236 = $561.80
Step 3: Subtract initial principal to isolate interest: 561.80 - 500 = $61.80
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Problem 3
A deposit of $2000 is made into an account earning 4% interest compounded annually. What is the total value of the investment after 2 years?
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Answer: $2163.20
Step 1: Growth multiplier = $$1 + 0.04 = 1.04$$. Formula: $$A = 2000 \times (1.04)^2$$
Step 2: Solve the exponent component: $$1.04 \times 1.04 = 1.0816$$
Step 3: Multiply by the principal: 2000 × 1.0816 = $2163.20
Calculate step-by-step using exponents for the time periods, adding interest back before computing the next year.