We can solve this problem by determining how many cupcakes were eaten first, then finding the fraction of cupcakes left. Here's how to approach it step by step:

1. Fraction of Cupcakes Eaten:

• 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.

We need to find the total fraction eaten by adding these two fractions:

$\text{Total fraction eaten} = \frac{2}{5} + \frac{3}{7}$

To add fractions, we need a common denominator. The least common denominator (LCD) of 5 and 7 is 35.

Now, convert both fractions:

$\frac{2}{5} = \frac{14}{35}, \quad \frac{3}{7} = \frac{15}{35}$

Now add them:

$\frac{14}{35} + \frac{15}{35} = \frac{29}{35}$

So, $\frac{29}{35}$ of the cupcakes were eaten.

2. Fraction of Cupcakes Left:

The fraction of cupcakes left is:

$\text{Fraction left} = 1 - \frac{29}{35} = \frac{35}{35} - \frac{29}{35} = \frac{6}{35}$

So, $\frac{6}{35}$ of the cupcakes are left.

3. Possible Total Number of Cupcakes:

To estimate how many cupcakes there might have been, we assume the total number of cupcakes is a multiple of 35, since the fraction $\frac{6}{35}$ represents the cupcakes left. For instance:

• If there were 35 cupcakes in total, then $\frac{6}{35} \times 35 = 6$ cupcakes would be left.
• If there were 70 cupcakes, $\frac{6}{35} \times 70 = 12$ cupcakes would be left.

Thus, the total number of cupcakes could have been any multiple of 35, such as 35, 70, 105, etc.

Would you like more details on any of the steps? Here are some related questions:

1. How do you find the least common denominator between two fractions?
2. What are some real-life examples of adding fractions?
3. How do you subtract fractions with unlike denominators?
4. What does it mean to multiply fractions by whole numbers?
5. Can you explain more examples of fractions in different contexts?

Tip: When adding fractions, always find the least common denominator first to simplify the process!