Let's define some variables to create a system of equations to represent this situation.
Let:
�
x be the time (in hours) the tortoise runs.
�
y be the time (in hours) the hare runs.
We know that the tortoise's speed is 2 ½ mph, which is equivalent to
2.5
2.5 mph, and the hare's speed is 8 ½ mph, which is equivalent to
8.5
8.5 mph.
The distance both the tortoise and the hare run is equal to their respective speeds multiplied by the time they run:
For the tortoise: Distance =
2.5
�
2.5x miles
For the hare: Distance =
8.5
�
8.5y miles
According to the problem, the hare reaches the finish line 45 minutes (3/4 of an hour) before the tortoise, which means the hare's running time (
�
y) is 45 minutes less than the tortoise's running time (
�
x). In terms of hours, this is
3
/
4
3/4 hours:
�
=
�
−
3
/
4
y=x−3/4
Now, we can set up the equations:
Equation 1:
�
�
�
�
�
�
�
�
tortoise
=
2.5
�
Distance
tortoise
=2.5x
Equation 2:
�
�
�
�
�
�
�
�
hare
=
8.5
�
Distance
hare
=8.5y
Equation 3:
�
=
�
−
3
/
4
y=x−3/4
To find how long the tortoise ran, we need to solve this system of equations. We'll substitute the expression for
�
y from Equation 3 into Equation 2:
�
�
�
�
�
�
�
�
hare
=
8.5
(
�
−
3
/
4
)
Distance
hare
=8.5(x−3/4)
Now, we'll set the distances equal since they both reach the finish line:
2.5
�
=
8.5
(
�
−
3
/
4
)
2.5x=8.5(x−3/4)
Now, let's solve for
�
x:
2.5
�
=
8.5
�
−
6.375
2.5x=8.5x−6.375
Subtract
8.5
�
8.5x from both sides:
2.5
�
−
8.5
�
=
−
6.375
2.5x−8.5x=−6.375
Combine like terms:
−
6
�
=
−
6.375
−6x=−6.375
Now, divide both sides by
−
6
−6 to solve for
�
x:
�
=
−
6.375
−
6
≈
1.0625
x=
−6
−6.375
≈1.0625 hours
To express this time in minutes, multiply by 60 (since there are 60 minutes in an hour):
�
≈
1.0625
×
60
≈
63.75
x≈1.0625×60≈63.75 minutes
So, the tortoise ran for approximately 63.75 minutes, or 1 hour and 3 minutes.