During rush hour, Bill can drive 60 miles using the side roads in the same time that it takes to travel 30 miles on the freeway. If Bill's rate on the side roads is 7 mi/h faster than his rate on the freeway, find his rate on the side roads.
let bill drive 30 mile (1/2 time unit of sideroads, one time unit on freeway)
Distance are the same
distance sideroad= distance freeway
v*time= (v+ 7)time/2
or divide through by time
v= v/2+3.5
solve for v the speed on the freeway.
To find Bill's rate on the side roads, we can set up an equation using the given information.
Let's say Bill's rate on the freeway is v mph. Then his rate on the side roads would be (v + 7) mph, as it is stated that his rate on the side roads is 7 mph faster than his rate on the freeway.
We know that during rush hour, Bill can drive 60 miles using the side roads in the same time that it takes to travel 30 miles on the freeway.
So, we can set up the equation:
Distance on side roads = Distance on freeway
Rate on side roads * Time on side roads = Rate on freeway * Time on freeway
(v + 7) * Time on side roads = v * Time on freeway
Since we are given that the time on the side roads is half of the time on the freeway, we can substitute (Time on freeway / 2) for Time on side roads:
(v + 7) * (Time on freeway / 2) = v * Time on freeway
Now we can simplify the equation:
(v + 7) * (1/2) = v
Multiplying both sides by 2 to remove the fraction:
(v + 7) = 2v
Expanding the left side:
v + 7 = 2v
Subtracting v from both sides:
7 = v
So, Bill's rate on the side roads is 7 mph.
Thus, Bill's rate on the side roads is 7 mph faster than his rate on the freeway.