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Zorluk: OrtaPercent Change and Interest

An investor distributed a total principal of $20,000\$20,000 between two accounts, Account X and Account Y. Account X earns simple annual interest at a rate of 8%8\%, while Account Y earns compound annual interest at a rate of 10%10\% compounded annually. If no further deposits or withdrawals were made and the total interest earned from both accounts combined at the end of 22 years was $3,600\$3,600, how much money was initially invested in Account X?

  1. A
    $8,000\$8,000
  2. B
    $10,000\$10,000
  3. $12,000\$12,000Cevap
  4. D
    $14,000\$14,000
  5. E
    $15,000\$15,000

Cevap

$12,000\$12,000
Account X earns 16%16\% simple interest over 2 years (8%×28\% \times 2), and Account Y earns 21%21\% compound interest over 2 years ((1.10)2−1=0.21(1.10)^2 - 1 = 0.21). Letting the amount in Account X be PXP_X, the total interest equation is 0.16PX+0.21(20,000−PX)=3,6000.16 P_X + 0.21(20,000 - P_X) = 3,600. Expanding yields 4,200−0.05PX=3,6004,200 - 0.05 P_X = 3,600, which simplifies to 0.05PX=6000.05 P_X = 600, giving PX=12,000P_X = 12,000.

Adım Adım Çözüm

1
Express the interest rate for Account X over 2 years.
Account X earns simple interest at 8%8\% per year for 22 years, giving a total interest multiplier of 0.08×2=0.160.08 \times 2 = 0.16 or 16%16\%.
Simple interest accumulates linearly as Principal ×\times Rate ×\times Time.
2
Express the interest rate for Account Y over 2 years.
Account Y earns compound interest at 10%10\% per year compounded annually for 22 years. The total growth factor is (1+0.10)2=1.21(1 + 0.10)^2 = 1.21, meaning the interest earned is 1.21−1=0.211.21 - 1 = 0.21 or 21%21\% of the principal.
Compound interest multiplies the balance each period.
3
Set up an equation using the total principal of $20,000\$20,000 and total interest of $3,600\$3,600.
Let PXP_X be the principal in Account X. Then the principal in Account Y is 20,000−PX20,000 - P_X. The total interest equation is 0.16PX+0.21(20,000−PX)=3,6000.16 P_X + 0.21(20,000 - P_X) = 3,600.
The sum of interest from both accounts equals the total interest given.
4
Solve for PXP_X.
0.16PX+4,200−0.21PX=3,600  ⟹  −0.05PX=3,600−4,200  ⟹  −0.05PX=−600  ⟹  PX=12,0000.16 P_X + 4,200 - 0.21 P_X = 3,600 \implies -0.05 P_X = 3,600 - 4,200 \implies -0.05 P_X = -600 \implies P_X = 12,000.
Isolating the variable PXP_X gives the initial principal invested in Account X.

Anahtar Kavram

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Percent Change and Interest

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