We have two investments for Sherrie.
The first, an unknown amount X at 13% interest.
The second investment is a capital of $110 more than five times the amount X, invested 5 times within the year, at a rate of 8%.
The interest of the first investment, after one year (t=1), can be calcualted as:
[tex]I_1=r_1X=0.13X[/tex]The interest for the second investment can be calculated as:
[tex]\begin{gathered} FV=PV(1+\frac{r_2}{5})^5 \\ I_2=FV-PV=PV\lbrack(1+\frac{r^{}_2}{5})^5-1\rbrack \\ I_2=(110+5X)\lbrack(1+\frac{0.08}{5})^5-1\rbrack=(110+5X)\cdot(1\text{.}016^5-1) \\ I_2=(110+5X)\cdot(1.0826-1)=(110+5X)\cdot0.0826=9.09+0.413X \end{gathered}[/tex]We know that the total interest in the year is $1051.84.
Then, the value X can be calculated as:
[tex]\begin{gathered} I_1+I_2=0.13X+(9.09+0.413X)=1051.84 \\ 0.13X+0.413X=1051.84-9.09=1042.75 \\ X=\frac{1042.75}{0.543}=1920\text{.}35 \end{gathered}[/tex]At 13% rate, Sherrie invested $1920.35.
At 8% rate, Sherrie invested $9711.75.
[tex]5\cdot(1920.35)+110=9601.75+110=9711\text{.}75[/tex]