Alex deposited $$x$$ dollars into a new account that earned $$8$$ percent annual interest, compounded annually. One year

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Alex deposited $$x$$ dollars into a new account that earned $$8$$ percent annual interest, compounded annually. One year

by Gmat_mission » Sun Jun 06, 2021 10:42 am

00:00

A

B

C

D

E

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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. $$\dfrac{w}{1+1.08}$$

B. $$\dfrac{w}{1.08+1.16}$$

C. $$\dfrac{w}{1.16+1.24}$$

D. $$\dfrac{w}{1.08+1.08^2}$$

E. $$\dfrac{w}{1.08^2+1.08^2}$$

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Re: Alex deposited $$x$$ dollars into a new account that earned $$8$$ percent annual interest, compounded annually. One

by Ian Stewart » Tue Jun 08, 2021 5:03 am
In the first year, the initial deposit of $x earns 8% interest, so after one year, the account holds 1.08x dollars. Then an additional$x is deposited in the account, so the account now contains 1.08x + x = x(1.08 + 1) dollars.

This amount now earns 8% interest over the second year, so at the end of two years, the account holds (1.08)(x)(1.08 + 1) dollars. Since this amount is w, we can solve the following for x:

w = (1.08)(x)(1.08 + 1)
x = w/[(1.08)(1.08 + 1)]
x = w/(1.08^2 + 1.08)