To travel 175 miles, it takes Sue, riding a moped, 4 hours less time than it takes Doreen to travel 66 miles riding a bicycle. Sue travels 19 miles per hour faster than Doreen. Find the times and rates of both girls.
Let Doreen's speed = x and Sue's = x + 19.
Let y - 4 = Doreen's time and y = Sue's time
Speed = distance/time
x = 175/(y-4) (Doreen)
x + 19 = 66/y (Sue)
Substitute the value of x from the first equation into the second and solve.
I hope this helps. Thanks for asking.
To solve this problem, we can use a system of equations. Let's denote Sue's speed as S and Doreen's speed as D. We also need to find the times taken by each person, so let's denote Sue's time as t1 and Doreen's time as t2.
We are given three pieces of information:
1. To travel 175 miles, Sue takes 4 hours less time than Doreen takes to travel 66 miles. This can be written as:
175/S = 66/D - 4 (Equation 1)
2. Sue's speed is 19 miles per hour faster than Doreen's speed:
S = D + 19 (Equation 2)
3. We want to find the times and rates of both girls.
Now, let's solve the system of equations.
First, we can solve Equation 2 for D in terms of S:
D = S - 19
Next, substitute this into Equation 1:
175/S = 66/(S - 19) - 4
To simplify the equation, let's multiply everything by S(S - 19) to remove the denominators:
175(S - 19) = 66S - 4S(S - 19)
Expand and rearrange the equation:
175S - 3325 = 66S - 4S^2 + 76S
Rearrange it to create a quadratic equation:
4S^2 - 85S + 3325 = 0
Now, we can use the quadratic formula to solve for S:
S = (-b ± √(b^2 - 4ac)) / 2a
Applying it to our equation:
S = (-(-85) ± √((-85)^2 - 4 * 4 * 3325)) / (2 * 4)
S = (85 ± √(7225 - 53200)) / 8
S = (85 ± √(-45975)) / 8
Since the discriminant (√(7225 - 4 * 4 * 3325)) is negative, this means there are no real solutions for S. Therefore, there is no valid solution for this problem.
In conclusion, there are no valid values for Sue's and Doreen's speeds, as well as the times taken by each girl, that satisfy the given conditions.