Problem: Vivian and Noelle both leave the park at the same time, but in opposite directions. If Noelle travels
5 mph faster than Vivian and after 8 hours, they are 136 miles apart, how fast in mile per hour is each
traveling?
Let,
Rate of Noel = 𝑅𝑁
Rate of Vivien = 𝑅𝑉
Given:
𝑅𝑁 = 5 𝑚𝑝ℎ + 𝑅𝑉 = 5 + 𝑅𝑉
Distance between them after 8 hours (D) = 136 miles
Find:
𝑅𝑁 and 𝑅𝑉
Figure:
Their position after 8 hours:
Distance traveled by Noelle and Vivien after 8 hours were 𝑑𝑁 and 𝑑𝑉 respectively, therefore the given distance
of 136 miles that they were apart is also equal to 𝑑𝑁 + 𝑑𝑉 . Since they are moving in opposite direction their
distances adds up. This is the very first thing we should understand in this case, to move on.
𝐷 = 𝑑𝑁 + 𝑑𝑉
→ Equation 1
Also, their individual distances can be computed from this rate formula,
𝑅=
𝑑
𝑡
Thus, we can now solve for their individual distances both at time 𝑡 = 8, for they leave at the same time and
same point which is the park.
𝑑𝑁 = 𝑅𝑁 (𝑡) = (5 + 𝑅𝑉 )(8)
; where 𝑅𝑁 = 5 + 𝑅𝑉 from the given.
And, 𝑑𝑉 = 𝑅𝑉 (𝑡) = (𝑅𝑉 )(8)
Finally, from equation 1, substitute above values,
𝐷 = 𝑑𝑁 + 𝑑𝑉
136 = (5 + 𝑅𝑉 )(8) + (𝑅𝑉 )(8)
136 = 8(5) + 8𝑅𝑉 + 8𝑅𝑉
136 = 40 + 16𝑅𝑉
136 − 40 = 16𝑅𝑉
96 = 16𝑅𝑉
Dividing both side by 16,
96 16𝑅𝑉
=
16
16
𝑹𝑽 = 𝟔 𝒎𝒑𝒉
Substituting this value from our given,
𝑅𝑁 = 5 𝑚𝑝ℎ + 𝑅𝑉 = 5 + 6
𝑹𝑵 = 𝟏𝟏 𝒎𝒑𝒉