To solve this problem, we need to set up a system of equations based on the given information.
Let's assume Bella spends x hours using the jogging track and y hours renting a bike. According to the given information, the cost equation for Bella would be:
8x + 10y = 44 ---(Equation 1)
In the same way, let's assume Jamar spends x hours using the track and y hours renting a bike. The cost equation for Jamar would be:
4x + 5y = 22 ---(Equation 2)
Now, we can solve this system of equations to find the values of x and y.
To eliminate one variable, multiply Equation 1 by 2 and Equation 2 by -1:
16x + 20y = 88 ---(Equation 3)
-4x - 5y = -22 ---(Equation 4)
Adding Equation 3 and Equation 4 together:
(16x + 20y) + (-4x - 5y) = 88 + (-22)
12x + 15y = 66
Now, we can solve this equation for either of the variables. Let's solve it for x:
12x = 66 - 15y
x = (66 - 15y) / 12
Since we want to find the number of hours, x and y should be non-negative integers. We can try different values for y and see which ones yield integer values for x.
Let's start by checking y = 0:
x = (66 - 15(0)) / 12
x = 66 / 12
x = 5.5
Since x should be an integer, y = 0 is not a valid solution.
Let's try y = 1:
x = (66 - 15(1)) / 12
x = 51 / 12
x โ 4.25
Again, x is not an integer, which means y = 1 is not a valid solution.
Let's try y = 2:
x = (66 - 15(2)) / 12
x = 36 / 12
x = 3
Finally, we have found an integer solution. If Bella spends 3 hours on the track and 2 hours renting a bike, she will pay a total of $44.
Now that we have determined Bella's values, we can substitute these values into either Equation 1 or Equation 2 to find Jamar's values.
Let's use Equation 1:
8x + 10y = 44
8(3) + 10y = 44
24 + 10y = 44
10y = 44 - 24
10y = 20
y = 2
Therefore, Jamar spends 3 hours at the track, same as Bella, and also spends 2 hours renting a bike.
In summary, both Bella and Jamar spend 3 hours at the track.