One day students in school organised a exchange between them such that : $11$ strawberries change for $14$ raspberries, $22$ cherries change for $21$ raspberries, $10$ cherries change for $3$ bananas and $5$ pears for $2$ bananas. How many pears has Amila to give to get $7$ strawberries
1. Define the variables for each type of fruit:
   \[
   s = \text{strawberries}, \quad r = \text{raspberries}, \quad c = \text{cherries}, \quad b = \text{bananas}, \quad p = \text{pears}
   \]

2. Write down the given exchange rates:
   \[
   11s \to 14r
   \]
   \[
   22c \to 21r
   \]
   \[
   10c \to 3b
   \]
   \[
   5p \to 2b
   \]

3. From the first exchange rate, we can express strawberries in terms of raspberries:
   \[
   11s = 14r \implies s = \frac{14}{11}r
   \]

4. From the second exchange rate, we can express cherries in terms of raspberries:
   \[
   22c = 21r \implies c = \frac{21}{22}r
   \]

5. To find a common term, we can express both strawberries and cherries in terms of raspberries and then relate them:
   \[
   s = \frac{14}{11}r \quad \text{and} \quad c = \frac{21}{22}r
   \]

6. To eliminate raspberries, we can find a common multiple:
   \[
   33s = 42r \quad \text{and} \quad 44c = 42r
   \]
   Therefore:
   \[
   33s = 44c \implies 3s = 4c
   \]

7. From the third exchange rate, we can express cherries in terms of bananas:
   \[
   10c = 3b \implies c = \frac{3}{10}b
   \]

8. From the fourth exchange rate, we can express pears in terms of bananas:
   \[
   5p = 2b \implies p = \frac{2}{5}b
   \]

9. Substitute the expression for cherries in terms of bananas into the equation \(3s = 4c\):
   \[
   4c = 4 \left(\frac{3}{10}b\right) = \frac{12}{10}b = \frac{6}{5}b
   \]
   Therefore:
   \[
   3s = \frac{6}{5}b \implies s = \frac{2}{5}b
   \]

10. Now, substitute the expression for bananas in terms of pears:
    \[
    b = \frac{5}{2}p
    \]
    Therefore:
    \[
    s = \frac{2}{5} \left(\frac{5}{2}p\right) = p
    \]

11. Since \(s = p\), to get 7 strawberries, Amila needs to give:
    \[
    7s = 7p
    \]

Conclusion:
\[
\boxed{7 \ \text{pears}}
\]