Answer
$8.5$ hours; $7.5$ hours;
Work Step by Step
Let's note:
$x$=the time (in hours) the first person needs to mow the lawn alone
$y$=the time (in hours) the second person needs to mow the lawn alone
In one hour the first person mows $1/x$ from the lawn, while the second $1/y$, so when they work together in one hour they do $(1/x)+(1/y)$ which is $1/4$ from the lawn.
Write the two equations:
$$\begin{cases}
\dfrac{1}{x}+\dfrac{1}{y}&=\dfrac{1}{4}\\
x&=y-1.
\end{cases}$$
Multiply the first equation by $4xy$:
$$\begin{cases}
4x+4y&=xy\\
x&=y-1.
\end{cases}$$
Substitute $x=y-1$ in the first equation and solve for $y$ using the Quadratic Formula:
$$\begin{align*}
4(y-1)+4y&=(y-1)y\\
4y-4+4y&=y^2-y\\
8y-4&=y^2-y\\
0&=y^2-y-8y+4\\
y^2-9y+4&=0\\
y&=\dfrac{-(-9)\pm\sqrt{(-9)^2-4(1)(4)}}{2(1)}\\
&=\dfrac{9\pm 8.062}{2}\\
y_1&\approx 0.5\\
y_2&\approx 8.5.
\end{align*}$$
Because $y>1$, the only solution that fits is $y=8.5$.
Determine the time (in hours) the first person need to mow the lawn alone:
$$x=y-1=8.5-1=7.5.$$
