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Zorluk: ZorPercentages, 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%Cevap
  3. C
    44%
  4. D
    60%
  5. E
    84%

Cevap

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\%.

Adım Adım Çözüm

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).

Anahtar Kavram

Simple vs. Compound Interest Base Shift and Net Percent Change
Bu soruyu puanla

Konular

Percentages, Percent Change, and Interest

Bu konudan daha fazla soru