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.
