Dionne can run around a circular track in 120 seconds. Basha, running in the opposite direction as Dionne, meets Dionne every 48 seconds. Sandra running in the same directions as Basha, passes Basha every 240 seconds. How often does Sandra meet Dionne?
To solve this question, let's break it down step by step:
First, let's determine the speed of Dionne. We know she can complete one lap around the circular track in 120 seconds. Therefore, her speed is 1 lap per 120 seconds or 1/120 lap per second.
Next, let's find the speed of Basha. We are told that Basha meets Dionne every 48 seconds while running in the opposite direction. Since they are running in opposite directions, their relative speed is the sum of their individual speeds. Let's say Basha's speed is x laps per second. So, the relative speed of Basha and Dionne would be (x + 1/120) laps per second.
We also know that Basha completes one lap around the track when she meets Dionne every 48 seconds. Therefore, her speed is 1 lap per 48 seconds or 1/48 lap per second.
Using the relative speed equation, we can set up the following equation:
x + 1/120 = 1/48
To solve for x, we need to find a common denominator for the fractions on both sides of the equation. The common denominator is 240:
240x + 2 = 5
240x = 5 - 2
240x = 3
x = 3/240
Simplifying the fraction, we get x = 1/80.
So, Basha's speed is 1/80 lap per second.
Now let's find out how often Sandra meets Dionne. We are told that Sandra passes Basha every 240 seconds while running in the same direction as Basha. Since they are running in the same direction, their relative speed would be the difference of their individual speeds. Let's say Sandra's speed is y laps per second. So, the relative speed of Sandra and Basha would be (y - 1/80) laps per second.
We also know that Sandra completes one lap around the track when she passes Basha every 240 seconds. Therefore, her speed is 1 lap per 240 seconds or 1/240 lap per second.
Using the relative speed equation, we can set up the following equation:
y - 1/80 = 1/240
To solve for y, we need to find a common denominator for the fractions on both sides of the equation. The common denominator is 240:
240y - 3 = 1
240y = 1 + 3
240y = 4
y = 4/240
Simplifying the fraction, we get y = 1/60.
So, Sandra's speed is 1/60 lap per second.
Now we can determine how often Sandra meets Dionne. Since Dionne can complete one lap in 120 seconds, and Sandra's speed is 1/60 lap per second, we can calculate the frequency of their meetings by dividing the lap time by the lap speed:
Meeting frequency = 120 seconds / (1/60 lap per second)
Simplifying this expression, we get:
Meeting frequency = 120 seconds * (60 lap per second)
Meeting frequency = 7200 meetings per second
Therefore, Sandra meets Dionne every 7200 seconds.