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Difficulty: HardPercent Change and Interest

An investor deposits $5,000\$5,000 into Account B, which earns interest at a rate of 20%20\% per year compounded annually for 22 years. The same investor deposits another $5,000\$5,000 into Account A, which earns simple annual interest at a rate of r%r\% per year for 33 years. If the total interest earned from Account B exceeds the total interest earned from Account A by $400\$400, what is the value of rr?

Answer: 12 %

Answer

The annual simple interest rate r is 12.
To find rr, calculate the interest from each account. Account B grows compounded annually to $5,000×(1.20)2=$7,200\$5,000 \times (1.20)^2 = \$7,200, producing $2,200\$2,200 in interest. Account A produces 5,000×r100×3=150r5,000 \times \frac{r}{100} \times 3 = 150r in simple interest. The problem states that 2,200−150r=4002,200 - 150r = 400. Solving for rr gives 150r=1,800150r = 1,800, so r=12r = 12.

Step-by-Step Solution

1
Calculate the compound interest earned from Account B
Interest from Account B = $2,200
Using the compound interest formula A=P(1+i)nA = P(1 + i)^n, the balance after 2 years is $5,000×(1.20)2=$7,200\$5,000 \times (1.20)^2 = \$7,200. Subtracting the principal gives $7,200−$5,000=$2,200\$7,200 - \$5,000 = \$2,200.
2
Express the simple interest earned from Account A in terms of r
Interest from Account A = 150r
Simple interest is calculated as I=P×r100×t=5,000×r100×3=150rI = P \times \frac{r}{100} \times t = 5,000 \times \frac{r}{100} \times 3 = 150r.
3
Formulate and solve the linear equation relating the two interest amounts
r = 12
Subtracting the simple interest from the compound interest gives 2,200−150r=4002,200 - 150r = 400. Solving yields 150r=1,800150r = 1,800, which gives r=12r = 12.

Key Concept

Comparing simple interest and compound interest expressions to solve for an unknown rate
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Topics

Percent Change and Interest

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