Let's solve this 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.

2. Total fraction of cupcakes eaten: To find the total fraction of cupcakes eaten, we need to add $\frac{2}{5}$ and $\frac{3}{7}$.

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

   • Convert $\frac{2}{5}$ to have denominator 35:
     $\frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35}$
   • Convert $\frac{3}{7}$ to have denominator 35:
     $\frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35}$

   Now, add the two fractions: $\frac{14}{35} + \frac{15}{35} = \frac{29}{35}$

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

3. Fraction of cupcakes left: To find the fraction of cupcakes that are left, subtract the fraction eaten from 1 (since 1 represents the whole):

   $1 - \frac{29}{35} = \frac{35}{35} - \frac{29}{35} = \frac{6}{35}$

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

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How many cupcakes might there have been?

Let the total number of cupcakes be $N$. Since the fraction left is $\frac{6}{35}$, the number of cupcakes left is $\frac{6}{35} \times N$.

A plausible value for $N$ would be a multiple of 35, as the denominator of the fraction is 35. Some possibilities include:

• 35 cupcakes: 6 cupcakes left
• 70 cupcakes: 12 cupcakes left
• 105 cupcakes: 18 cupcakes left

Any multiple of 35 could work, depending on the total number of cupcakes.

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

1. If 12 cupcakes are left, how many were originally there?
2. What if the purple table group ate twice as many cupcakes as the green table group?
3. What fraction of cupcakes would be left if the green and purple groups ate $\frac{4}{7}$ instead of $\frac{3}{7}$?
4. How would the result change if the red and blue groups ate $\frac{3}{5}$ of the cupcakes instead?
5. Can you think of a situation where more than one group eats the same fraction of cupcakes?

Tip: Always look for the least common denominator when adding fractions with different denominators!