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Difficulty: MediumPercentages, Percent Change, and Interest

An investor divides a total initial investment of $10,000\$10,000 between Account XX and Account YY. Account XX earns simple annual interest at a rate of 6%6\%, while Account YY earns annual interest at a rate of 10%10\% compounded annually. If no additional deposits or withdrawals are made and the total interest earned from both accounts combined at the end of 22 years is $1,560\$1,560, how much money was originally invested in Account YY?

  1. A
    $3,500\$3,500
  2. $4,000\$4,000Answer
  3. C
    $4,500\$4,500
  4. D
    $6,000\$6,000
  5. E
    $6,400\$6,400

Answer

$4,000\$4,000
The correct answer of $4,000\$4,000 is determined by calculating the interest earned by each account over two years. Account X earns simple interest of 6%×2=12%6\% \times 2 = 12\%, producing interest equal to 0.12(10,000−PY)0.12(10,000 - P_Y). Account Y earns compound interest at 10%10\% annually, yielding an interest factor of 1.102−1=0.211.10^2 - 1 = 0.21, or 0.21PY0.21 P_Y. Adding the interest expressions yields 1,200+0.09PY=1,5601,200 + 0.09 P_Y = 1,560, which simplifies to 0.09PY=3600.09 P_Y = 360 and gives PY=$4,000P_Y = \$4,000.

Step-by-Step Solution

1
Define variables for the initial principal in each account
Let PYP_Y be the principal in Account YY. Then the principal in Account XX is 10,000−PY10,000 - P_Y.
The total investment across both accounts is $10,000\$10,000.
2
Calculate the interest earned by Account XX over 22 years
\text{Interest}_X = (10,000 - P_Y) \times 0.06 \times 2 = 0.12(10,000 - P_Y) = 1,200 - 0.12 P_Y.
Account XX earns simple annual interest of 6%6\% per year for 22 years.
3
Calculate the interest earned by Account YY over 22 years
\text{Interest}_Y = P_Y \times ((1 + 0.10)^2 - 1) = P_Y \times (1.21 - 1) = 0.21 P_Y.
Account YY earns interest compounded annually at 10%10\% per year for 22 years.
4
Formulate and solve the equation for total interest
(1,200 - 0.12 P_Y) + 0.21 P_Y = 1,560 \implies 1,200 + 0.09 P_Y = 1,560 \implies 0.09 P_Y = 360 \implies P_Y = 4,000.
The combined interest from both accounts equals $1,560\$1,560.

Key Concept

Simple vs Compound Interest over multiple time periods
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Topics

Percentages, Percent Change, and Interest

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