Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. When they met, Josh had ridden for twice the length of time as Mike and at 4/5 of Mike's rate. How many miles had Mike ridden when they met?
(a) 4
(b) 5
(c) 6
(d) 7
(e) 8
To solve this problem, we can use a system of equations.
Let's denote the rate at which Mike is riding as M miles/hour, and the time he rides as T hours. Therefore, the distance Mike has ridden is M * T.
From the given information, we know that when they met, Josh had ridden for twice the length of time as Mike, so Josh's time is 2T hours. We also know that Josh's rate is 4/5 of Mike's rate, so Josh's rate is (4/5)M miles/hour.
Since the total distance they covered is 13 miles, we can write the equation:
Mike's distance + Josh's distance = 13
M * T + ((4/5)M) * (2T) = 13
Simplifying this equation, we get:
MT + (8/5)MT = 13
Adding the fractional terms, we get:
(13/5)MT = 13
Dividing both sides by (13/5), we find:
MT = 5
Dividing both sides of the equation by M, we find:
T = 5/M
Now, looking at the answer choices, we see that the distance Mike has ridden is M * T. Therefore, we need to find a value of M that satisfies the equation 5/M.
Let's check each answer choice:
(a) 4 * 5/4 = 5 (Not the correct answer)
(b) 5 * 5/5 = 5 (Not the correct answer)
(c) 6 * 5/6 = 5 (Not the correct answer)
(d) 7 * 5/7 = 5 (Not the correct answer)
(e) 8 * 5/8 = 5 (Correct! Mike has ridden 5 miles when they met)
Therefore, the correct answer is (b) 5. Mike had ridden 5 miles when they met.
Vj = 0.8 Vm
Josh rides for time t at speed Vj
Mike rides for time t/2 at speed Vm
Vm*t/2 + Vj*t = 13
Vm*t/2 + 0.8 Vm*t = 13
1.3 Vm t = 13
Vm t = 10
The distance Mike travelled is
(1/2) Vm t = 5 miles
You cannot solve for Vm and t separately, but you don't have to to get the answer.