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].
Answer (1)
Define the variable d as the distance Ario has run from his starting point when Miguel starts.
Set up the equation 22 − d d = 4 1 based on the given ratio.
Solve for d to find d = 4.4 meters.
Calculate Ario's position from the start of the pool: 3 + 4.4 = 7.4 meters. The final answer is 7.4
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 end. Miguel gives Ario a head start and starts when the ratio of Ario's completed distance to his remaining distance is 1:4. We need to find Ario's position (meter mark) when Miguel starts the race.
Define Variables and Total Distance Let d be the distance Ario has run from his starting point when Miguel begins the race. The total length of the pool available for the race is 25 − 3 = 22 meters. When Miguel starts, Ario has run d meters, and the remaining distance for Ario to run is 22 − d meters.
Set up the Ratio Equation We are given that the ratio of Ario's completed distance to his remaining distance is 1:4. Therefore, we can write the equation: 22 − d d = 4 1
Solve for d Now, we solve for d : 4 d = 22 − d 5 d = 22 d = 5 22 = 4.4 This means Ario has run 4.4 meters from his starting point.
Calculate Ario's Position Since Ario started 3 meters from the beginning of the pool, his position when Miguel starts is 3 + d = 3 + 4.4 = 7.4 meters.
Final Answer Therefore, Ario is at the 7.4-meter mark when Miguel starts the race.
Examples
Understanding relative distances and ratios is crucial in various real-life scenarios, such as planning routes or managing resources. For instance, imagine you're organizing a charity run where participants get a head start based on age or ability. By calculating the ratio of the distance already covered to the remaining distance, you can determine when to signal the start for other runners to ensure a fair and engaging race. This problem demonstrates how mathematical ratios help in creating balanced and equitable conditions in competitive events.
