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Difficulty: HardPercentages, Percent Change, and Interest

An investment fund allocated its total initial capital equally into two separate accounts, Account X and Account Y. Account X earns simple annual interest at a rate of r%r\%, while Account Y earns compound annual interest at the same rate of r%r\%, compounded annually. At the end of 2 years, the total interest earned by Account Y exceeded the total interest earned by Account X by an amount equal to 4%4\% of the initial amount placed in Account X. What was the net percentage increase in the fund's total combined capital over the 2-year period?

  1. A
    40%
  2. 42%Answer
  3. C
    44%
  4. D
    60%
  5. E
    84%

Answer

42%
The correct answer of 42% is found by recognizing that the 2-year difference between compound interest and simple interest at rate r%r\% equals P(r/100)2P(r/100)^2. Setting this difference to 0.04P0.04P yields r=20%r = 20\%. At r=20%r = 20\%, Account X produces 0.40P0.40P in simple interest, while Account Y produces 0.44P0.44P in compound interest. Together, they generate 0.84P0.84P in interest on a total initial investment of 2P2P, resulting in a net fund increase of (0.84P/2P)×100%=42%(0.84P / 2P) \times 100\% = 42\%.

Step-by-Step Solution

1
Set up expressions for interest earned by each account after 2 years
Let PP be the initial capital in Account X (so Account Y also starts with PP, making total initial capital 2P2P). Simple interest for Account X is IX=P⋅2r100=0.02rPI_X = P \cdot \frac{2r}{100} = 0.02r P. Compound interest for Account Y is IY=P[(1+r100)2−1]=P(2r100+r210000)I_Y = P\left[\left(1 + \frac{r}{100}\right)^2 - 1\right] = P\left(\frac{2r}{100} + \frac{r^2}{10000}\right).
Establishes the interest expressions based on simple vs. compound interest formulas over a 2-year period.
2
Calculate the annual interest rate rr
Difference in interest: IY−IX=P(r210000)I_Y - I_X = P\left(\frac{r^2}{10000}\right). Given this equals 4%4\% of PP (0.04P0.04P), we have r210000=0.04  ⟹  r2=400  ⟹  r=20%\frac{r^2}{10000} = 0.04 \implies r^2 = 400 \implies r = 20\%.
Isolates the interest rate rr using the difference between compound and simple interest.
3
Determine total interest earned across both accounts
For Account X: IX=0.02(20)P=0.40PI_X = 0.02(20)P = 0.40P. For Account Y: IY=0.40P+0.04P=0.44PI_Y = 0.40P + 0.04P = 0.44P. Total interest earned = 0.40P+0.44P=0.84P0.40P + 0.44P = 0.84P.
Sums the returns of both portfolio components.
4
Compute the net percentage increase of the total fund
Total initial capital = 2P2P. Net percentage increase = Total InterestTotal Initial Capital×100%=0.84P2P×100%=42%\frac{\text{Total Interest}}{\text{Total Initial Capital}} \times 100\% = \frac{0.84P}{2P} \times 100\% = 42\%.
Calculates the overall percent change using the correct part-to-whole base (total initial investment 2P2P).

Key Concept

Simple vs. Compound Interest Base Shift and Net Percent Change
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Topics

Percentages, Percent Change, and Interest

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