https://file.aiquickdraw.com/mathbot/file/1725644243972vwrvi0yf.jpg

I have opened the image. I will now analyze and extract the mathematical content for further explanation.It appears the image contains a series of math word problems. Let me transcribe and solve each of them:

### Problem 1:
A dog weighs 0.75 pounds more than twice the cat's weight. The total weight of both animals is 12.25 pounds. 
Let \( x \) be the cat's weight. The equation for this problem is:

\[
\text{Dog's weight} = 2x + 0.75
\]
\[
\text{Total weight of both} = x + (2x + 0.75) = 12.25
\]

Simplifying:

\[
x + 2x + 0.75 = 12.25
\]
\[
3x + 0.75 = 12.25
\]
\[
3x = 12.25 - 0.75
\]
\[
3x = 11.5
\]
\[
x = \frac{11.5}{3} = 3.83 \text{ pounds (cat's weight)}
\]

Now, the dog's weight:
\[
2x + 0.75 = 2(3.83) + 0.75 = 7.66 + 0.75 = 8.41 \text{ pounds (dog's weight)}
\]

So, the cat weighs 3.83 pounds, and the dog weighs 8.41 pounds.

---

### Problem 2:
Carlos can finish a task in 5 hours. Juan can finish it in 3 hours. If they work together, how long will it take to complete the task?

Let \( t \) be the time it takes for both to finish the task working together.

Carlos' rate is \( \frac{1}{5} \) tasks per hour, and Juan's rate is \( \frac{1}{3} \) tasks per hour. Together:

\[
\frac{1}{t} = \frac{1}{5} + \frac{1}{3}
\]

Finding a common denominator:
\[
\frac{1}{t} = \frac{3}{15} + \frac{5}{15} = \frac{8}{15}
\]

Thus:
\[
t = \frac{15}{8} = 1.875 \text{ hours}
\]

It will take Carlos and Juan 1.875 hours, or 1 hour and 52.5 minutes, to finish the task together.

---

Would you like more details or any clarifications? 

Here are some related questions to consider:

1. How would the equation for Problem 1 change if the dog weighed 1 pound more than twice the cat's weight?
2. How does the concept of work rates apply to larger tasks with more people?
3. What if Carlos and Juan worked at different times during the day, how would that impact the total work time?
4. How can you graph the relationship between the cat's weight and the dog's weight in Problem 1?
5. What if the combined weight of the dog and cat was 20 pounds instead of 12.25?

**Tip**: When working with rate problems, always try to convert the problem into equations based on units (like hours per task).

A dog weighs 0.75 pounds more than twice the cat's weight. The total weight of both animals is 12.25 pounds. How long will it take Carlos and Juan to complete a task if they work together, given Carlos can finish it in 5 hours and Juan in 3 hours?

Explore how to solve a real-world problem involving the weight of a dog and cat using linear equations. Additionally, learn how to calculate the time it takes for two people working together to complete a task using work rate formulas. This content is suitable for students in Grades 6-8.

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