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At what meter mark will Ano be when Miguel starts the race? Round to the nearest tenth.

Given:
[tex]\[
x = \left(\frac{m}{m+n}\right)\left(x_2 - x_1\right) + x_1
\][/tex]

Miguel and his brother Ano are both standing 3 meters from one side of a 25-meter pool when they decide to race. Miguel offers Ano a head start. Miguel says he will start when the ratio of Ano's completed meters to Ano'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 :

GinnyAnswer
Sure, let's break down the steps to determine at what meter mark Ano will be when Miguel starts the race.

### Step 1: Identify the Given Values

We are given the following information:

- [tex]\( x_1 = 3 \)[/tex] meters (the starting point for both Miguel and Ano)
- [tex]\( x_2 = 25 \)[/tex] meters (the total length of the pool)
- The ratio of Ano's completed meters to remaining meters is [tex]\( 1:4 \)[/tex]

### Step 2: Understand the Ratio

The ratio [tex]\( 1:4 \)[/tex] means that for every 1 meter Ano has completed, he has 4 meters remaining. We can denote these two parts of the ratio as [tex]\( m = 1 \)[/tex] and [tex]\( n = 4 \)[/tex].

### Step 3: Plug Values into the Formula

The formula provided is:
[tex]\[ x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1 \][/tex]

### Step 4: Substitute the Values

Substitute [tex]\( m \)[/tex], [tex]\( n \)[/tex], [tex]\( x_1 \)[/tex], and [tex]\( x_2 \)[/tex] into the formula:
[tex]\[ x = \left(\frac{1}{1+4}\right)(25 - 3) + 3 \][/tex]

### Step 5: Simplify Inside the Parentheses

Calculate [tex]\( 1 + 4 \)[/tex]:
[tex]\[ 1 + 4 = 5 \][/tex]

Then calculate [tex]\( 25 - 3 \)[/tex]:
[tex]\[ 25 - 3 = 22 \][/tex]

### Step 6: Continue Substitution

Now substitute these results back into the formula:
[tex]\[ x = \left(\frac{1}{5}\right) \times 22 + 3 \][/tex]

### Step 7: Calculate the Fraction

[tex]\[ \left(\frac{1}{5}\right) \times 22 = \frac{22}{5} = 4.4 \][/tex]

### Step 8: Add the Initial Position [tex]\( x_1 \)[/tex]

[tex]\[ x = 4.4 + 3 \][/tex]

### Step 9: Final Calculation

[tex]\[ x = 7.4 \][/tex]

### Conclusion

Therefore, Ano will be at the 7.4 meter mark when Miguel starts the race. This is rounded to the nearest tenth.

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