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The tortoise and the hare agree to a re-match. As they begin the race, the tortoise plods along slowly but confidently at 2 ½ mph. The hare zips ahead at 8 ½ mph, determined not to repeat his original mistake…and he doesn’t! The hare never slows down and reaches the finish line 45 minutes (3/4 of an hour) before the tortoise. How long did the tortoise run? Write a system of equations with x and y.

User EKelvin
by
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1 Answer

3 votes

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.

User Pim Reijersen
by
7.8k points