Meena owns a bottle cap collection. While on a vacation, she finds a large number of bottle caps, increasing her collection size by $40\%$. Later on her same vacation, she decides that she does not like some of the bottle caps, so she gives away $20\%$ of her current collection. Suppose that Meena owns $21$ more bottle caps after her vacation than before her vacation. How many bottle caps did Meena have before her vacation?
1. Let \( x \) be the number of bottle caps that Meena owned before her vacation.
2. During her vacation, Meena increases her collection by \( 40\% \). Therefore, the number of bottle caps after the increase is:
   \[
   x + 0.4x = 1.4x
   \]
3. Later, she gives away \( 20\% \) of her current collection. The number of bottle caps she gives away is:
   \[
   0.2 \times 1.4x = 0.28x
   \]
4. The number of bottle caps remaining after giving away \( 20\% \) is:
   \[
   1.4x - 0.28x = 1.12x
   \]
5. According to the problem, Meena has 21 more bottle caps after her vacation than before. Therefore, we set up the equation:
   \[
   1.12x - x = 21
   \]
6. Simplify the equation:
   \[
   0.12x = 21
   \]
7. Solve for \( x \):
   \[
   x = \frac{21}{0.12} = 175
   \]

The final answer is \( \boxed{175} \)