You're walking down an infinite street, and you notice that six out of every seven cats on a sidewalk are followed by a dog, while one out of every four dogs is followed by a cat. What proportion of animals on the sidewalk are dogs? Let c and d, respectively, be the proportion of cats and dogs on the sidewalk. Note that of all the cats, 6/7 ​ are followed by a dog. Iterate this type of logic.

To solve this problem, we need to find the proportion of animals on the sidewalk that are dogs. Let’s break this down step by step using the variables \( c \) and \( d \), which represent the proportion of cats and dogs, respectively, on the sidewalk.

### Step 1: Initial Conditions
We are given the following information:

1. **Six out of every seven cats are followed by a dog**. This means that the number of cats followed by a dog is \( \frac{6}{7}c \).

2. **One out of every four dogs is followed by a cat**. This means that the number of dogs followed by a cat is \( \frac{1}{4}d \).

Our goal is to find the proportion of dogs, \( d \), among all animals.

### Step 2: Relationships Between Cats and Dogs

The total number of animals is the sum of the proportions of cats and dogs, which gives:
\[
c + d = 1
\]

Next, we need to express \( c \) and \( d \) in terms of each other using the information provided.

### Step 3: Establishing the Balance Between Cats and Dogs

#### For Cats:
We are told that six out of every seven cats are followed by a dog. So, the number of cats followed by dogs is \( \frac{6}{7}c \). This means the proportion of dogs must account for these dogs that follow cats. In other words, the number of dogs is related to \( \frac{6}{7}c \).

#### For Dogs:
We are also told that one out of every four dogs is followed by a cat. So, the number of dogs that follow cats is \( \frac{1}{4}d \), which must correspond to the proportion of cats that are followed by dogs. In other words, \( \frac{1}{4}d \) is related to the proportion of cats.

### Step 4: Solving for Proportions

From these two relationships, we can set up an equation. The number of dogs following cats is equal to the number of cats following dogs:
\[
\frac{6}{7}c = \frac{1}{4}d
\]
We can solve for \( d \) in terms of \( c \):
\[
d = \frac{24}{7}c
\]

### Step 5: Using the Total Proportion

We know that the total proportion of animals is 1, so:
\[
c + d = 1
\]
Substitute \( d = \frac{24}{7}c \) into the equation:
\[
c + \frac{24}{7}c = 1
\]
Multiply through by 7 to eliminate the fraction:
\[
7c + 24c = 7
\]
\[
31c = 7
\]
\[
c = \frac{7}{31}
\]

### Step 6: Finding \( d \)

Now that we have \( c = \frac{7}{31} \), we can find \( d \) using \( d = 1 - c \):
\[
d = 1 - \frac{7}{31} = \frac{24}{31}
\]

### Final Answer:
The proportion of animals that are dogs is \( \frac{24}{31} \).

Would you like further clarification or more details on any step?

### Related Questions:
1. What is the proportion of cats on the sidewalk?
2. How does the solution change if only half of the cats are followed by dogs?
3. What happens if the ratio of dogs followed by cats changes to 1/3 instead of 1/4?
4. How would you represent this problem in terms of a system of linear equations?
5. What is the significance of using proportions in solving this type of problem?

### Tip:
When solving problems involving proportions, try setting up relationships and equations based on the given ratios, and look for a way to express variables in terms of one another.

You're walking down an infinite street, and you notice that six out of every seven cats on a sidewalk are followed by a dog, while one out of every four dogs is followed by a cat. What proportion of animals on the sidewalk are dogs? Let c and d, respectively, be the proportion of cats and dogs on the sidewalk. Note that of all the cats, 6/7 ​are followed by a dog. Iterate this type of logic.

Explore how to find the proportion of dogs on a sidewalk where six out of every seven cats are followed by dogs, and one out of every four dogs is followed by a cat. This algebraic problem demonstrates proportional reasoning and equation balancing, suitable for high school students.

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