ANSWER
[tex]\$92,592.59[/tex]
EXPLANATION
We want to find how much Juan needs to invest now to meet his target.
To do this, we have to apply the formula for the amount for a quarterly compounded interest:
[tex]A=P(1+\frac{r}{4})^{4t}[/tex]
where 4 represents 4 quarters in a year
P = principal or initial amount invested
r = interest rate
A = amount after t years
t = number of years
Therefore, substituting the given values into the equation, we have that:
[tex]\begin{gathered} 250000=P(1+\frac{\frac{5}{100}}{4})^{4\cdot20} \\ 250000=P(1+\frac{5}{400})^{80} \\ 250000=P(1+0.0125)^{80} \\ 250000=P(1.0125)^{80} \\ 250000=P\cdot2.70 \end{gathered}[/tex]
Solve for P by dividing both sides by 2.70:
[tex]\begin{gathered} P=\frac{250000}{2.7} \\ P=\$92,592.59 \end{gathered}[/tex]
That is the amount that he needs to invest now.
