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Zorluk: Çok zorPercentages, Percent Change, and Interest

At the beginning of Year 1, an investor divided a sum of money between two portfolio accounts, Account X and Account Y, such that the initial balance of Account X was 25%25\% greater than the initial balance of Account Y.

Over a two-year period:
- Account X earned compound interest at a constant annual rate of 20%20\% per year.
- Account Y decreased in value by 10%10\% during Year 1, and then increased in value by r%r\% during Year 2.

If the total combined value of both accounts at the end of Year 2 was 32%32\% greater than the total combined initial balance at the beginning of Year 1, what is the value of rr?

  1. A
    2222
  2. B
    2525
  3. C
    2727
  4. 3030Cevap
  5. E
    3636

Cevap

The value of rr is 3030.
The correct value is 3030. Account X starts at 125%125\% of Account Y's initial value (125125 vs 100100, totaling 225225). Compounding Account X at 20%20\% per year for 2 years yields 125×1.202=180125 \times 1.20^2 = 180. Account Y drops 10%10\% to 9090 in Year 1. For the overall total to reach 225×1.32=297225 \times 1.32 = 297, Account Y must reach 297−180=117297 - 180 = 117 at the end of Year 2. The percentage increase from 9090 to 117117 is 117−9090×100%=2790×100%=30%\frac{117 - 90}{90} \times 100\% = \frac{27}{90} \times 100\% = 30\%.

Adım Adım Çözüm

1
Define initial account balances using a suitable variable base
Let initial balance of Account Y be Y0=100Y_0 = 100 units. Initial balance of Account X is X0=1.25×100=125X_0 = 1.25 \times 100 = 125 units. Combined initial balance is T0=125+100=225T_0 = 125 + 100 = 225 units.
Establishing a standard numerical base simplifies tracking multi-account percentage changes.
2
Calculate the value of Account X at the end of Year 2
X2=125×(1+0.20)2=125×1.44=180X_2 = 125 \times (1 + 0.20)^2 = 125 \times 1.44 = 180 units.
Account X compounds annually at 20%20\% for 2 full years.
3
Express the value of Account Y at the end of Year 2 in terms of rr
After Year 1: Y1=100×(1−0.10)=90Y_1 = 100 \times (1 - 0.10) = 90 units. After Year 2: Y2=90×(1+r100)Y_2 = 90 \times \left(1 + \frac{r}{100}\right) units.
Account Y first loses 10%10\% of its initial value, creating a new base of 9090 units for the Year 2 growth rate.
4
Determine the required total combined final value
T2=225×(1+0.32)=225×1.32=297T_2 = 225 \times (1 + 0.32) = 225 \times 1.32 = 297 units.
The overall portfolio value grew by 32%32\% relative to the initial combined balance of 225225 units.
5
Set up and solve the combined equation for rr
180+90(1+r100)=297  ⟹  90(1+r100)=117  ⟹  1+r100=1.30  ⟹  r=30180 + 90 \left(1 + \frac{r}{100}\right) = 297 \implies 90 \left(1 + \frac{r}{100}\right) = 117 \implies 1 + \frac{r}{100} = 1.30 \implies r = 30.
Equating the sum of individual final account balances to the target portfolio balance yields the exact value of rr.

Anahtar Kavram

Successive percent changes on shifting bases combined with compound interest calculations across multiple accounts.
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Konular

Percentages, Percent Change, and Interest

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