Answered

Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.

Jeff can weed the garden twice as fast as his sister Julia. Together, they can weed the garden in 3 hours. How long would it take each of them working alone?

Which of the following equations could be used to solve this problem?

A. [tex]\left(\frac{1}{x}\right)+\left(\frac{1}{2x}\right)=3[/tex]

B. [tex]\left(\frac{1}{x}\right) \cdot 3+\left(\frac{1}{2x}\right) \cdot 3=1[/tex]

C. [tex]\left(\frac{1}{x}\right)+\left(\frac{1}{2x}\right)=1[/tex]

D. [tex]\left(\frac{1}{x}\right) \cdot 6+\left(\frac{1}{2x}\right) \cdot 3=1[/tex]

Sagot :

GinnyAnswer
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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.