To determine the ratio of tops to dresses to shoes sold, we begin by noting the fractions given for each category of sales:

• 1/3 of sales were for tops.
• 1/5 of sales were for dresses.
• The remaining sales were for shoes.

First, let's find the fraction of sales for shoes by subtracting the fractions for tops and dresses from 1 (the total sales):

$\text{Shoes} = 1 - \left(\frac{1}{3} + \frac{1}{5}\right)$

To subtract these fractions, we need a common denominator. The least common denominator (LCD) of 3 and 5 is 15, so let's convert the fractions:

$\frac{1}{3} = \frac{5}{15}, \quad \frac{1}{5} = \frac{3}{15}$

Now, add them:

$\frac{1}{3} + \frac{1}{5} = \frac{5}{15} + \frac{3}{15} = \frac{8}{15}$

Now subtract this sum from 1:

$\text{Shoes} = 1 - \frac{8}{15} = \frac{15}{15} - \frac{8}{15} = \frac{7}{15}$

So, the fraction of sales for shoes is 7/15.

Now, the sales distribution is as follows:

• Tops: $\frac{1}{3}$
• Dresses: $\frac{1}{5}$
• Shoes: $\frac{7}{15}$

To express these fractions as a ratio, we need to find a common denominator for all three fractions. The least common denominator of 3, 5, and 15 is 15. Let's express each fraction with a denominator of 15:

$\frac{1}{3} = \frac{5}{15}, \quad \frac{1}{5} = \frac{3}{15}, \quad \frac{7}{15} = \frac{7}{15}$

Now the ratio of tops to dresses to shoes is:

$5 : 3 : 7$

Thus, the simplest ratio of tops to dresses to shoes sold is 5:3:7.

Would you like more details or have any further questions?

Here are 5 related questions:

1. How would the ratio change if half of the sales were for tops?
2. What if 40% of the sales were for shoes, how would that affect the ratio?
3. Can you express the same ratio using percentages instead of fractions?
4. What is the total proportion of sales for tops and dresses together?
5. If 105 total items were sold, how many were shoes based on the ratio?

Tip: To simplify a ratio, find a common denominator for all the fractions involved!