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Zorluk: Çok zorPercent Change and Interest

An investor allocated a total principal of $25,000\$25,000 between two savings accounts, Account X and Account Y. Account X earns simple interest at an annual rate of 8%8\%, while Account Y earns interest at an annual rate of 10%10\%, compounded annually. If after 22 years the total interest earned from Account Y exceeds the total interest earned from Account X by $1,550\$1,550, what was the amount, in dollars, invested in Account Y?

Cevap: 15000 dollars

Cevap

The amount invested in Account Y was $15,000 dollars.
Let PYP_Y be the principal invested in Account Y. Since the total investment is $25,000\$25,000, the principal invested in Account X is 25,000−PY25,000 - P_Y. Account X earns simple interest over 2 years equal to (25,000−PY)×0.08×2=4,000−0.16PY(25,000 - P_Y) \times 0.08 \times 2 = 4,000 - 0.16 P_Y. Account Y earns compound interest over 2 years equal to PY×((1+0.10)2−1)=0.21PYP_Y \times ((1 + 0.10)^2 - 1) = 0.21 P_Y. The problem states that the interest from Account Y exceeds that from Account X by $1,550\$1,550, giving the equation 0.21PY−(4,000−0.16PY)=1,5500.21 P_Y - (4,000 - 0.16 P_Y) = 1,550. Combining like terms yields 0.37PY=5,5500.37 P_Y = 5,550, which simplifies to PY=15,000P_Y = 15,000.

Adım Adım Çözüm

1
Express the principal of Account X in terms of Account Y
Principal of Account X = 25,000−PY25,000 - P_Y
The total sum invested across both accounts is $25,000\$25,000.
2
Calculate the interest earned from Account X over 2 years
IX=4,000−0.16PYI_X = 4,000 - 0.16 P_Y
Simple interest is calculated as I=P×r×t=(25,000−PY)×0.08×2I = P \times r \times t = (25,000 - P_Y) \times 0.08 \times 2.
3
Calculate the compound interest earned from Account Y over 2 years
IY=0.21PYI_Y = 0.21 P_Y
The compound multiplier for 2 years at 10%10\% is (1.10)2=1.21(1.10)^2 = 1.21, giving a interest percentage of 1.21−1=0.211.21 - 1 = 0.21 or 21%21\%.
4
Set up and simplify the equation for the difference in interest
0.37PY=5,5500.37 P_Y = 5,550
Subtracting IXI_X from IYI_Y yields 0.21PY−(4,000−0.16PY)=1,5500.21 P_Y - (4,000 - 0.16 P_Y) = 1,550, which simplifies to 0.37PY−4,000=1,5500.37 P_Y - 4,000 = 1,550.
5
Solve for PYP_Y
PY=15,000P_Y = 15,000
Dividing 5,5505,550 by 0.370.37 gives 15,00015,000.

Anahtar Kavram

Combining Simple and Compound Interest Linear Equations
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Konular

Percent Change and Interest

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