Let's solve this problem step by step by defining variables and forming equations based on the given information.

Step 1: Define the Variables

Let:

• $C$ represent the number of cats.
• $D$ represent the number of dogs.

We are told that the rescue center has two-thirds as many dogs as cats. This gives us the relationship: $D = \frac{2}{3}C$

Step 2: Calculate the Total Food Costs for the Animals

• Each dog is fed 2 cans of food per day, and dog food costs $0.68 per can. Therefore, the daily cost for one dog is: $2 \times 0.68 = 1.36 \, \text{dollars/day}.$ For a 30-day month, the total food cost per dog is: $30 \times 1.36 = 40.8 \, \text{dollars/month}.$

• Each cat is also fed 2 cans of food per day, and cat food costs $0.65 per can. Therefore, the daily cost for one cat is: $2 \times 0.65 = 1.30 \, \text{dollars/day}.$ For a 30-day month, the total food cost per cat is: $30 \times 1.30 = 39.0 \, \text{dollars/month}.$

Step 3: Write the Total Budget Equation

The total food budget is $2581.80 for the month. The total cost for all the dogs and cats must add up to this amount. Therefore, the equation for the total cost becomes: $40.8D + 39.0C = 2581.80$

Step 4: Substitute $D = \frac{2}{3}C$ into the Budget Equation

Now substitute $D = \frac{2}{3}C$ into the equation: $40.8 \times \frac{2}{3}C + 39.0C = 2581.80$ Simplify: $27.2C + 39.0C = 2581.80$ $66.2C = 2581.80$

Step 5: Solve for $C$

Solve for $C$ (the number of cats): $C = \frac{2581.80}{66.2} \approx 39$

Step 6: Solve for $D$

Now that we know $C = 39$, use the relationship $D = \frac{2}{3}C$ to find the number of dogs: $D = \frac{2}{3} \times 39 = 26$

Final Answer:

There are 26 dogs at the rescue center.

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Would you like any additional details or have any questions?

Here are five related questions:

1. How would the budget change if the cost of cat food increased by $0.05 per can?
2. What if the number of dogs was one-half of the number of cats instead?
3. How would the monthly food cost change if each animal ate 3 cans per day instead of 2?
4. What is the cost per animal per day based on this budget?
5. If the budget increased by 10%, how many more dogs or cats could the center feed?

Tip: When solving word problems, define variables and set up equations based on relationships described in the problem.