Answer
Answer: Jogging speed = 6 miles/hour.
Work Step by Step
Let the time taken by the jogger going to the park be = $x$ hours.
Thus, she takes 1 hour more ($x+1$ hours) to return home from the longer route.
The rate is constant, so it does not matter what value we take.
So, let the rate be = $r$.
Thus, while going to the park,
Distance = 12 miles, time = $x$ hours.
Thus Distance = Rate $\times$ Time = $r(x)$
Thus 12 = $r(x)$
$\frac{12}{r}$ = $x$
x = $\frac{12}{r}$
And while returning ,
Distance = 18 miles, time = $(x+1)$ hours
Thus 18 = $r(x+1)$
$\frac{18}{r}$ = $x+1$
$x$ = $\frac{18}{r}-1$ = $\frac{18-r}{r}$
Thus,
$x$ = $\frac{12}{r}$ = $\frac{18-r}{r}$ hours.
Thus:
$\frac{12}{r}$ = $\frac{18-r}{r}$
Multiplying both sides by $r$,
$\frac{12}{r}$($r$) = $\frac{18-r}{r}$$(r)$ ....(Since $\frac{r}{r}=1$)
Thus $12=18-r$
Thus, adding $r$ on both sides,
$12+r=18-r+r$ = 18
Thus, subtracting 12 from both sides,
$12+r-12$ = $18-12$ = 6
$r$ = 6
Thus $r$ = 6 miles/hour
Jogging speed = 6 miles/hour.
