How long had each been driving?

Question

Thaddeus and Ian start at the same location and drive in opposite​ directions, but leave at different times. When they are 345 miles​ apart, their combined travel time is 8 hours. If Thaddeus drives at a rate of 40 miles per hour and Ian drives at a rate of 45 miles per​ hour, how long had each been​ driving?

Answer 1

Suppose Thaddeus drove for x hours, then Ian drove for 8-x hours

Set up the equation as 40 mph for x hours plus 45 mph for 8-x hours is a total of 345 miles

40x+45(8-x)=345

40x+360-45x=345

-5x=-15

x=3

Then Thaddeus drove for 3 hours and Ian drove for 8-3=5 hours

Answer 2

The total distance driven by both is 345 miles. Let that distance = X

Let The distance driven by Thaddeus be X_a and the distance driven by Ian be X_b

Then the total Distance X = 345 miles = X_a + X_b ---Eq 1

The total time driven by both is 8 hrs and let that time be T

Let the time driven by Thaddeus be T_a and the time driven by Ian be T_b

Then the total time driven by both T = 8 hrs = T_a + T_b ---Eq 2

The distance X_a = V_a T_a , where V_a is the speed of Thaddeus given by 40 mile/hr, Then X_a = 40 T_a

The distance X_b = V_b T_b , where V_b is the speed of Ian given by 45 mile/hr, Then X_b = 45 T_b

Then in Eq1 , 345 = 40 T_a+ 45 T_b , Now you have two equations two unknowns with T_a, and T_b as unknowns

345 = 40 T_a+ 45 T_b ...Eq 1 solve for T_a = 345 - 45 T_b / 40

8 = T_a + T_b ---Eq 2 solve for T_a = 8 - T_b

Equate T_a from both equations , 345 - 45 T_b / 40 = 8 -T_b Cross multiply, 345 - 45 T_b = 40( 8 -T_b)

345 - 45 T_b = 320 - 40 T_b , and 5 T_b = 25 , T_b = 5 hrs, and T_a = 8 - 5 = 3 hrs

Answer:

Thaddeus has been driving for 3 hrs, and Ian for 5 hrs