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Alex deposited x dollars into a new account

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Alex deposited x dollars into a new account

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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

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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)
Since 8% interest is earned each year, the amount in the account after the first year = 1.08x.
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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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)
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:

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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scott@targettestprep.com



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Quote:
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)
Hi rsarashi,
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.

Hope it helps.
I am available if you'd like any follow up.

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