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Yumi and Juan pick up litter along the highway at the same rate. Yumi picks up litter for 3 miles and takes 2 hours longer than Juan, who picks up litter for 2 miles. If Juan spent [tex]\( x \)[/tex] hours picking up litter, the following equation models the situation:

[tex]\[
\frac{2}{x} = \frac{3}{x+2}
\][/tex]

How long did Yumi spend picking up litter?

A. 1 hour 20 minutes
B. 3 hours 20 minutes
C. 4 hours
D. 6 hours



Answer :

GinnyAnswer
To find out how long Yumi spent picking up litter, we need to solve the given equation:
[tex]\[ \frac{2}{x} = \frac{3}{x + 2} \][/tex]

### Step-by-Step Solution:

1. Set up the Equation:
[tex]\[ \frac{2}{x} = \frac{3}{x + 2} \][/tex]

2. Cross-Multiply to Eliminate the Fractions:
[tex]\[ 2(x + 2) = 3x \][/tex]

3. Distribute and Simplify:
[tex]\[ 2x + 4 = 3x \][/tex]

4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 4 = 3x - 2x \][/tex]
[tex]\[ 4 = x \][/tex]
So, Juan spent [tex]\( x \)[/tex] hours picking up litter, which means Juan spent 4 hours.

5. Find How Long Yumi Spent Picking Up Litter:
Since Yumi took 2 hours longer than Juan:
[tex]\[ x + 2 \][/tex]
Substituting [tex]\( x = 4 \)[/tex]:
[tex]\[ 4 + 2 = 6 \][/tex]

Therefore, Yumi spent [tex]\( 6 \)[/tex] hours picking up litter.

So, the correct answer is:
[tex]\[ \boxed{6 \text{ hours}} \][/tex]

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