At a recent meeting, shoe styles were noticed. For every person wearing sneakers, there were two wearing dress shoes. For every two people wearing sandals, there were three wearing dress shoes. What is the ratio of sandals to sneakers?

To solve this, let’s define variables and set up the relationships given in the problem:

1. **Let \( s \)** represent the number of people wearing sneakers.
2. **Let \( d \)** represent the number of people wearing dress shoes.
3. **Let \( a \)** represent the number of people wearing sandals.

The problem provides the following relationships:

1. For every person wearing sneakers, there are two people wearing dress shoes:
   \[
   d = 2s
   \]

2. For every two people wearing sandals, there are three people wearing dress shoes:
   \[
   \frac{a}{2} = \frac{d}{3}
   \]
   Cross-multiplying to clear the fractions gives:
   \[
   3a = 2d
   \]

### Step 1: Substitute for \( d \) in terms of \( s \)

From the first equation, we know:
\[
d = 2s
\]

### Step 2: Substitute \( d = 2s \) into \( 3a = 2d \)

Using the second equation \( 3a = 2d \) and substituting \( d = 2s \):
\[
3a = 2(2s)
\]
\[
3a = 4s
\]

### Step 3: Solve for the ratio of \( a \) to \( s \)

Divide both sides by 3:
\[
a = \frac{4}{3}s
\]

This means the ratio of sandals to sneakers, \( \frac{a}{s} \), is:
\[
\frac{a}{s} = \frac{4}{3}
\]

### Conclusion
The ratio of sandals to sneakers is **4:3**.

Would you like further details or have any questions?

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Here are some related questions:

1. If there were 12 people wearing dress shoes, how many people were wearing sneakers and sandals?
2. What would the ratio of dress shoes to sneakers be if there were 10 people wearing sneakers?
3. How would the ratios change if there were equal numbers of people wearing each shoe style?
4. How can we determine the total number of people at the meeting if we know one of the shoe counts?
5. If 15 people wore sandals, how many people wore sneakers and dress shoes?

**Tip:** When solving ratio problems, identify all relationships and try to express everything in terms of one variable for easier calculation.

Explore how to find the ratio of sandals to sneakers when given proportions of dress shoes to other shoe types. This problem involves setting up ratios and solving for unknowns in a real-life context. Ideal for students in Grades 6-8 looking to strengthen their understanding of ratios and basic algebra.

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