Let's break down the problem step by step.

1. Dividing the pizza

Lucas divides the pizza into three equal pieces, so Teddy's portion is:

$\frac{1}{3} \text{ of the pizza.}$

2. Teddy's lunch consumption

Teddy eats $\frac{5}{8}$ of his piece for lunch. This means the fraction of his piece that remains after lunch is:

$1 - \frac{5}{8} = \frac{3}{8}.$

3. Teddy's dinner consumption

For dinner, Teddy eats $\frac{2}{5}$ of the remaining fraction of his piece. The remaining fraction is $\frac{3}{8}$, so the amount he eats for dinner is:

$\frac{2}{5} \times \frac{3}{8} = \frac{6}{40} = \frac{3}{20}.$

4. Fraction of the whole pizza

Since Teddy's original portion was $\frac{1}{3}$ of the pizza, the fraction of the whole pizza he ate for dinner is:

$\frac{3}{20} \times \frac{1}{3} = \frac{3}{60} = \frac{1}{20}.$

Conclusion

Teddy ate $\frac{1}{20}$ of the whole pizza for dinner.

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Would you like further details or clarifications?

Here are some related questions you might find interesting:

1. What fraction of the pizza did Teddy eat in total (lunch and dinner)?
2. If Lucas's other friend also ate part of his piece, how much of the pizza would be left?
3. How much of the pizza did Teddy leave uneaten after lunch and dinner combined?
4. What if Teddy had eaten a different fraction for dinner, say $\frac{1}{3}$ of what remained?
5. How could this problem change if Lucas divided the pizza into more than three pieces?

Tip: Always check intermediate steps in fraction calculations to avoid errors in the final result.