Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually. One year later Alex deposited an additional x dollars into the account. If there were no other transactions and if the account contained w dollars at the end of two years, which of the following expresses x in terms of w ?
A. w/(1+1.08)
B. w/(1.08+1.16)
C. w/(1.16+1.24)
D. w/(1.08+1.08^2)
E. w/(1.08^2+1.08^2)
D
Alex deposited x dollars into a new account
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Since 8% interest is earned each year, the amount in the account after the first year = 1.08x.rsarashi wrote:Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually. One year later Alex deposited an additional x dollars into the account. If there were no other transactions and if the account contained w dollars at the end of two years, which of the following expresses x in terms of w ?
A. w/(1+1.08)
B. w/(1.08+1.16)
C. w/(1.16+1.24)
D. w/(1.08+1.08^2)
E. w/(1.08^2+1.08^2)
After Alex deposits an additional x dollars, the amount in the account = 1.08x + x.
After an additional 8% is earned in the second year, the amount in the account = w = 1.08[1.08x + x)] = (1.08)²x + 1.08x
Since w = (1.08)²x + 1.08x, we get:
w = x(1.08² + 1.08)
x = w/(1.08² + 1.08).
The correct answer is D.
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We start by determining the new value of the x dollars Alex deposited into his account that earned 8 percent annual interest. At the end of the first year the amount of money in the account was 1.08x and then he added another x dollars to the account, so the account then had a total value of 1.08x + x dollars. The 1.08x + x dollars earned another 8 percent interest for the year. Thus, the total value of his account at the end of the second year is:rsarashi wrote:Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually. One year later Alex deposited an additional x dollars into the account. If there were no other transactions and if the account contained w dollars at the end of two years, which of the following expresses x in terms of w ?
A. w/(1+1.08)
B. w/(1.08+1.16)
C. w/(1.16+1.24)
D. w/(1.08+1.08^2)
E. w/(1.08^2+1.08^2)
1.08(1.08x + x) = (1.08^2)x + 1.08x
Since the new total value is equal to w, we can set up the following equation:
w = (1.08^2)x + 1.08x
Now we must get x in terms of w.
w = x(1.08^2 + 1.08)
w/(1.08^2 + 1.08) = x
Answer: D
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Hi rsarashi,Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually. One year later Alex deposited an additional x dollars into the account. If there were no other transactions and if the account contained w dollars at the end of two years, which of the following expresses x in terms of w ?
A. w/(1+1.08)
B. w/(1.08+1.16)
C. w/(1.16+1.24)
D. w/(1.08+1.08^2)
E. w/(1.08^2+1.08^2)
Let's take a look at your question.
Amount Alex deposited in account = x dollars
Annual interest earned after one year = (8%) x = 0.08x
Total amount in account after one year = (x + 0.08x) = 1.08x
After one year, Alex deposited additional x dollars.
Total amount after depositing x dollars = 1.08x + x
interest after second year = (8%)(1.08x + x) = (0.08)(1.08x + x)
Total amount after second year :
$$=\left(1.08x+x\right)+0.08\left(1.08x+x\right)$$
$$=\left(1.08x+x\right)\left(1+0.08\right)$$
$$=\left(1.08x+x\right)\left(1.08\right)$$
$$=\left(1.08\right)^2x+1.08x$$
$$=x\left[\left(1.08\right)^2+1.08\right]$$
Let w be the total amount in the account, then
$$w=x\left[\left(1.08\right)^2+1.08\right]$$
$$x=\frac{w}{1.08^2+1.08}$$
Therefore, Option D is correct.
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