To solve the problem, we need to find how long it takes both Julia and Jeff to weed the garden when working alone. Here is the step-by-step approach to solve it.
Let's define the variables:
- Let [tex]\( x \)[/tex] be the time it takes Julia to weed the garden alone.
- Jeff can weed the garden twice as fast as Julia, so it takes him [tex]\( x/2 \)[/tex] time.
Since we know that together they can weed the garden in 3 hours, we can set up an equation based on their combined work rates:
1. The amount of garden Julia can weed in one hour is [tex]\( \frac{1}{x} \)[/tex] (since she can complete the garden in [tex]\( x \)[/tex] hours).
2. The amount of garden Jeff can weed in one hour is [tex]\( \frac{1}{x/2} = \frac{2}{x} \)[/tex] (since he can complete it in [tex]\( x/2 \)[/tex] hours).
Combining their work rates, we have:
[tex]\[ \frac{1}{x} + \frac{2}{x} \][/tex]
Since they can complete the garden in 3 hours working together, we multiply their combined work rate by 3 and set it equal to the entire garden (1 complete garden):
[tex]\[ 3 \left( \frac{1}{x} + \frac{2}{x} \right) = 1 \][/tex]
Simplify the equation inside the parentheses:
[tex]\[ 3 \left( \frac{1}{x} + \frac{2}{x} \right) = 3 \left( \frac{3}{x} \right) = \frac{3 \cdot 3}{x} = \frac{9}{x} \][/tex]
So, we have:
[tex]\[ \frac{9}{x} = 1 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 9 \][/tex]
This means Julia takes 9 hours to weed the garden alone.
To find how long it takes Jeff to weed the garden alone, we use the fact that Jeff weeds twice as fast as Julia:
[tex]\[ \text{Jeff's time} = \frac{x}{2} = \frac{9}{2} = 4.5 \, \text{hours} \][/tex]
Thus, it takes Julia 9 hours and Jeff 4.5 hours to weed the garden alone.
Among the given options, the equation that correctly represents the problem is:
[tex]\[ \left( \frac{1}{x} \right) \cdot 3 + \left( \frac{1}{2x} \right) \cdot 3 = 1 \][/tex]
Which simplifies to:
[tex]\[ \left( \frac{1}{x} \right) + \left( \frac{2}{x} \right) = 1 \quad \text{(after dividing both sides by 3)} \][/tex]
Hence, the correct equation to use is:
[tex]\[ \left( \frac{1}{x} \cdot 3 + \frac{1}{2x} \cdot 3 \right) = 1 \][/tex]
