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 (2)
Define x as the distance Ario has completed when Miguel starts.
Express the remaining distance as 22 − x .
Set up the ratio equation: 22 − x x = 4 1 .
Solve for x : x = 4.4 meters.
Calculate Ario's position: 3 + 4.4 = 7.4 meters.
Ario is at 7.4 meters when Miguel starts.
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 begins when the ratio of Ario's completed distance to his remaining distance is 1:4. We need to find Ario's position (in meters from the starting side of the pool) when Miguel starts the race.
Define Variables Let x be the distance Ario has completed from his starting point when Miguel starts the race. The total available distance for Ario is 25 − 3 = 22 meters. Therefore, the remaining distance for Ario is 22 − x meters.
Set up the Equation The problem states that the ratio of Ario's completed distance to his remaining distance is 1:4. We can write this as: 22 − x x = 4 1
Solve for x Now, we solve for x : 4 x = 22 − x 5 x = 22 x = 5 22 = 4.4 So, Ario has completed 4.4 meters from his starting point when Miguel starts the race.
Calculate Ario's Position Since Ario started 3 meters from one end of the pool, his position from that end when Miguel starts is 3 + x = 3 + 4.4 = 7.4 meters.
Final Answer Therefore, Ario will be 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, if you're organizing a relay race, knowing how to calculate head starts based on speed ratios ensures fairness. Similarly, in project management, understanding the ratio of completed tasks to remaining tasks helps in tracking progress and making informed decisions. This problem demonstrates how mathematical ratios can be applied to create equitable conditions or assess progress in different contexts.
In this problem, we calculated that Ario will be at 7.4 meters when Miguel starts the race. This was determined by solving the ratio condition set between Ario's completed distance and his remaining distance. The chosen answer is B. 7.4 meters.
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