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Question

Difficulty: MediumPercent Change and Interest

An investor deposited a principal sum of money into an account that earns simple annual interest at a rate of r%r\%. At the end of 44 years, the total accumulated balance in the account (principal plus interest) was $7,200\$7,200. If the account had instead earned simple annual interest at a rate of (r+2)%(r + 2)\% over the same 44-year period, the total accumulated balance would have been $7,680\$7,680. What was the original principal amount deposited, in dollars?

Answer: 6000 dollars

Answer

The original principal amount deposited was 6,0006,000 dollars.
The difference in accumulated balances over 44 years is $7,680−$7,200=$480\$7,680 - \$7,200 = \$480. Since simple interest is calculated strictly on the original principal PP, an annual rate increase of 2%2\% yields an additional 2%2\% of PP each year. Over 44 years, this total rate increase is 4×2%=8%4 \times 2\% = 8\%. Therefore, 8%8\% of PP equals $480\$480, giving 0.08P=4800.08P = 480, which yields P=6,000P = 6,000 dollars.

Step-by-Step Solution

1
Calculate the difference between the two final balances.
$7,680−$7,200=$480\$7,680 - \$7,200 = \$480
The additional $480\$480 represents the total extra simple interest earned over 44 years due to the higher interest rate.
2
Determine the cumulative percentage rate increase over the 4-year period.
4 years×2% per year=8%4 \text{ years} \times 2\% \text{ per year} = 8\%
Simple interest is calculated solely on the original principal for each year.
3
Set up and solve the equation for the principal amount PP.
8% of P=$480  ⟹  0.08P=480  ⟹  P=6,0008\% \text{ of } P = \$480 \implies 0.08P = 480 \implies P = 6,000
Dividing the additional dollar amount by the additional percentage yields the original base principal.

Key Concept

Simple Interest and Rate Sensitivity
Estimated Time:1m 30s
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Topics

Percent Change and Interest

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