Examkin

Soru

Zorluk: Çok zorPercent Change and Interest

An investor deposits a principal amount into a savings account that earns interest at a constant annual compound rate of r%r\%. At the end of the first year, immediately after annual interest is credited, the investor withdraws 20%20\% of the interest earned during that first year, leaving the original principal and all remaining interest in the account. At the end of the second year, the interest earned in the second year alone is 10%10\% greater than the interest earned in the first year alone. What is the value of rr?

  1. A
    10.0%10.0\%
  2. 12.5%12.5\%Cevap
  3. C
    15.0%15.0\%
  4. D
    25.0%25.0\%
  5. E
    37.5%37.5\%

Cevap

12.5%
The option specifying 12.5% is correct. Let PP be the initial principal. The interest earned in the first year is I1=PrI_1 = P r. After withdrawing 20%20\% of this interest, 80%80\% of the interest remains in the account, making the new principal for Year 2 equal to P+0.8Pr=P(1+0.8r)P + 0.8Pr = P(1 + 0.8r). The interest earned in Year 2 is I2=P(1+0.8r)rI_2 = P(1 + 0.8r)r. Since Year 2 interest is 10%10\% greater than Year 1 interest, I2=1.10I1I_2 = 1.10 I_1. Equating the two expressions gives P(1+0.8r)r=1.10PrP(1 + 0.8r)r = 1.10 Pr. Dividing both sides by PrPr yields 1+0.8r=1.101 + 0.8r = 1.10, which simplifies to 0.8r=0.100.8r = 0.10, so r=0.125r = 0.125 or 12.5%12.5\%.

Adım Adım Çözüm

1
Express the interest earned in Year 1 in terms of principal PP and interest rate rr.
Year 1 interest I1=P×rI_1 = P \times r.
Simple/compound interest earned over the first single compounding period on initial principal PP is P×rP \times r.
2
Determine the account balance at the start of Year 2 after the partial interest withdrawal.
Balance at start of Year 2 = P+0.80(P×r)=P(1+0.8r)P + 0.80(P \times r) = P(1 + 0.8r).
The investor keeps the principal PP and retains 80%80\% of Year 1 interest (I1−0.20I1=0.80I1I_1 - 0.20 I_1 = 0.80 I_1).
3
Express the interest earned in Year 2 (I2I_2) and set up the equation I2=1.10×I1I_2 = 1.10 \times I_1.
P(1+0.8r)×r=1.10×(P×r)P(1 + 0.8r) \times r = 1.10 \times (P \times r).
Year 2 interest is calculated on the updated balance at rate rr, and is given as 10%10\% greater than Year 1 interest.
4
Simplify the equation to solve for rr.
1+0.8r=1.10  ⟹  0.8r=0.10  ⟹  r=0.100.80=0.125=12.5%1 + 0.8r = 1.10 \implies 0.8r = 0.10 \implies r = \frac{0.10}{0.80} = 0.125 = 12.5\%.
Divide both sides by the non-zero quantity P×rP \times r, then solve linear equation for rr.

Anahtar Kavram

Percent Change and Compound Interest Base Tracking
Tahmini Süre:2m 0s
Bu soruyu puanla

Konular

Percent Change and Interest

Bu konudan daha fazla soru