Question

Catherine puts $\$1,100$ in an investment account that she expects will make $5\%$ interest for each three month period. However, after a year she realises she was wrong about the interest rate and she has $\$50$ less than she expected. Assuming the interest rate the account earns is constant, which of the following equations expresses the total amount of money, $x$, she will have after $t$ years using the actual rate?

• A. $x=1,100(1.04{)}^{4t}$
• B. $x=1,100(1.05{)}^{4t-50}$
• C. $x=1,100(1.04{)}^{t/3}$
• D. $x=1,100(1.035{)}^{4t}$

Answer 1

The correct option is A $x=1,100(1.04{)}^{4t}$

$\Rightarrow$ The formula for compound interest is $A=P{(1+\frac{r}{100})}^t$

$\Rightarrow$ where $P$ is the starting principal, $r$ is the rate expressed as a decimal, and $t$ is the number of times the interest is compounded.

$\Rightarrow$ Catherine received less than $5\%$ interest, so you can eliminate $\left(B\right)$ because $1.05=1+0.05$, indicating she was receiving $5\%$ interest. You can also eliminate $\left(C\right)$ because over the course of a year the interest is compounded $4$ times, not $\frac{1}{3}$ of a time.

$\Rightarrow$ Because Melanie invested $\$1,100$ at what she thought was $5\%$ compounded $4$ times, she expected $1,100(1+0.05{)}^4=\$1,337.06$ after a year.

$\Rightarrow$ Instead, she has $1,337.06-50=\$1,287.06$ after one year.

$\Rightarrow$ Because $t$ is in years in the answer choices, make $t=1$ in $\left(A\right)$ and $\left(D\right)$ and eliminate any choice which does not equal $\$1,287.06$. Only $\left(A\right)$ works.

Therefore, the answer is $x=1,100(1.04{)}^{4t}$.