Let's break down the problem step-by-step to find out at what meter mark Ario will be when Miguel starts the race, given the ratio of Ario's completed meters to Ario's remaining meters is [tex]\(1:4\)[/tex].
### Step-by-Step Solution:
#### Step 1: Identify the Given Information
- Ario and Miguel are both starting at 3 meters from one side of a 25-meter pool.
- The total length of the pool is 25 meters.
- The starting point (initial position) of both Ario and Miguel is [tex]\( x_1 = 3 \)[/tex] meters.
- The goal (end position) is [tex]\( x_2 = 25 \)[/tex] meters.
- The ratio of the completed meters to the remaining meters when Miguel starts the race is [tex]\( 1 : 4 \)[/tex].
#### Step 2: Understand the Given Ratio
The ratio [tex]\( 1:4 \)[/tex] means that for every 1 meter Ario has completed, there are 4 meters remaining.
Let [tex]\( x \)[/tex] be the distance Ario has traveled when Miguel starts the race.
- Completed meters: [tex]\( x - x_1 \)[/tex]
- Remaining meters: [tex]\( x_2 - x \)[/tex]
Thus, the ratio translates to:
[tex]\[ \frac{x - x_1}{x_2 - x} = \frac{1}{4} \][/tex]
#### Step 3: Set Up the Equation Using the Ratio
Substitute [tex]\( x_1 = 3 \)[/tex] meters and [tex]\( x_2 = 25 \)[/tex] meters into the equation:
[tex]\[ \frac{x - 3}{25 - x} = \frac{1}{4} \][/tex]
Multiply both sides by [tex]\( 4(25 - x) \)[/tex] to clear the fraction:
[tex]\[ 4(x - 3) = 25 - x \][/tex]
#### Step 4: Solve for [tex]\( x \)[/tex]
Expand and simplify the equation:
[tex]\[ 4x - 12 = 25 - x \][/tex]
Combine like terms:
[tex]\[ 4x + x = 25 + 12 \][/tex]
[tex]\[ 5x = 37 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{37}{5} \][/tex]
[tex]\[ x = 7.4 \][/tex]
#### Step 5: Interpret the Result
The calculated [tex]\( x \)[/tex] value indicates the meter mark at which Miguel will start the race. Therefore, when Ario is at 7.4 meters, Miguel will start the race.
### Conclusion
The meter mark at which Miguel will start the race when Ario has achieved a [tex]\(1:4\)[/tex] ratio of completed distance to remaining distance is:
[tex]\[ \boxed{7.4} \][/tex] meters.
