Question

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. $\frac{w}{1+1.08}$
• B. $\frac{w}{1.08+1.16}$
• C. $\frac{w}{1.16+1.24}$
• D. $\frac{w}{1.08+{\left(1.08\right)}^2}$
• E. $\frac{w}{{\left(1.08\right)}^2+{\left(1.08\right)}^3}$

Answer 1

Answer: (D)

$\frac{w}{1.08+{\left(1.08\right)}^2}$

Amount Alex deposited in account = $x$ dollars.

$\Rightarrow$ Annual interest earned after one year = $8\%ofx=0.08x$

$\Rightarrow$ Total amount in account after one year = $(x+0.08x)=1.08x$

$\Rightarrow$ After one year, Alex deposited additional $x$ dollars.

$\Rightarrow$ Total amount after depositing $x$ dollars = $1.08x+x$

$\Rightarrow$ Interest after second year = $8\%of(1.08x+x)=0.08(1.08x+x)$

$\Rightarrow$ Total amount after second year :

$\Rightarrow$ $(1.08x+x)+0.08(1.08x+x)$

$\Rightarrow$ $(1.08x+x)(1+0.08)$

$\Rightarrow$ $(1.08x+x)\left(1.08\right)$

$\Rightarrow$ $x[1.08+(1.08{)}^2]$

$\Rightarrow$ Let $w$ be the total amount in the account, then

$\Rightarrow$ $w=x[1.08+(1.08{)}^2]$

$\therefore$ $x=\frac{w}{1.08+(1.08{)}^2}$