At what meter mark will Ario be when Miguel starts the race? Round to the nearest tenth.
[tex]x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1[/tex]
Miquel and his brother Ario are both standing 3 meters from one side of a 25-meter pool when they decide to race. Miguel offers Ario a head start. Miguel says he will start when the ratio of Ario's completed meters to Ario's remaining meters is [tex]$1: 4$[/tex].
A. 4.4 meters
B. 7.4 meters
C. 17.6 meters
D. 20.6 meters
Answer (1)
Define d as the distance Ario has covered when Miguel starts.
Set up the ratio of completed distance to remaining distance: 22 − d d = 4 1 .
Solve the equation for d : d = 5 22 = 4.4 .
Ario will be at the 4.4 meter mark when Miguel starts the race.
Explanation
Problem Analysis Let's analyze the problem. Miguel and Ario are racing in a 25-meter pool, but they start 3 meters from one side. Miguel gives Ario a head start and begins when the ratio of Ario's completed distance to his remaining distance is 1:4. We need to find the exact meter mark Ario is at when Miguel starts the race.
Defining Variables and Total Distance Let d be the distance Ario has covered from his starting point when Miguel starts the race. Since they start 3 meters from one end of the 25-meter pool, the total available distance for Ario is 25 − 3 = 22 meters. Therefore, the remaining distance for Ario when Miguel starts is 22 − d meters.
Setting up the Ratio According to the problem, the ratio of Ario's completed distance to his remaining distance is 1:4. This can be written as: 22 − d d = 4 1
Solving for d Now, we solve the equation for d : 4 d = 22 − d 5 d = 22 d = 5 22 d = 4.4
Determining Ario's Position The distance d represents the meter mark Ario is at from his starting point when Miguel begins the race. Since Ario started 3 meters from the edge, Ario is at 4.4 meters from his starting point. Therefore, Ario is at the 3 + 4.4 = 7.4 meter mark from the edge of the pool. However, the question asks at what meter mark will Ario be when Miguel starts the race. Since Ario started at the 3-meter mark, the distance Ario covered is 4.4 meters. Thus, Ario will be at the 4.4 meter mark from his starting point when Miguel starts the race.
Final Answer Therefore, Ario will be at the 4.4 meter mark when Miguel starts the race, rounded to the nearest tenth.
Examples
Imagine two friends, Alice and Bob, are running a race on a track. Alice, being faster, decides to give Bob a head start. Alice starts running only when Bob has covered a certain distance such that the ratio of the distance Bob has run to the distance he has left is 1:4. This problem helps determine exactly where Bob is on the track when Alice starts running, ensuring a fair and exciting race. Understanding ratios and distances in this context can be applied to various real-life scenarios, such as planning routes, managing resources, or coordinating events.
