Let's solve the problem step by step:

1. Determine the fraction eaten by the groups:

   • The red and blue table groups ate $\frac{2}{5}$ of the cupcakes.
   • The green and purple table groups ate $\frac{3}{7}$ of the cupcakes.

2. Add the fractions to find out how much was eaten in total: $\frac{2}{5} + \frac{3}{7}$ To add these, we need a common denominator. The least common denominator between 5 and 7 is 35. Let's rewrite the fractions: $\frac{2}{5} = \frac{14}{35}, \quad \frac{3}{7} = \frac{15}{35}$ Now, add the fractions: $\frac{14}{35} + \frac{15}{35} = \frac{29}{35}$ So, $\frac{29}{35}$ of the cupcakes were eaten.

3. Find the fraction of cupcakes that were left: The total is 1 (representing all the cupcakes), so the fraction left is: $1 - \frac{29}{35} = \frac{35}{35} - \frac{29}{35} = \frac{6}{35}$ Therefore, $\frac{6}{35}$ of the cupcakes were left.

4. How many cupcakes might there have been?: Since the fractions suggest the total number of cupcakes could be a multiple of 35 (to match the denominator), possible numbers for the total cupcakes include 35, 70, 105, etc.

For example, if there were 35 cupcakes in total:

• The red and blue groups would have eaten $\frac{2}{5} \times 35 = 14$ cupcakes.
• The green and purple groups would have eaten $\frac{3}{7} \times 35 = 15$ cupcakes.
• That would leave $35 - 14 - 15 = 6$ cupcakes, which matches $\frac{6}{35}$.

Summary:

• The fraction of cupcakes left is $\frac{6}{35}$.
• The total number of cupcakes might be 35, 70, 105, etc.

Would you like more details on any part of the solution?
Here are five questions to deepen your understanding:

1. How do you find the least common denominator of two fractions?
2. Can you think of other methods to subtract fractions?
3. How could we verify the total number of cupcakes given the fractions?
4. What if the groups ate unequal amounts of cupcakes?
5. Can the total number of cupcakes be something other than a multiple of 35?

Tip: When adding or subtracting fractions, always ensure they have a common denominator!