To solve the expression \( \frac{9}{8} \div 7 \div 6 \times \frac{14}{3} \times 20 \times \frac{22}{11} \), we will follow the order of operations (PEMDAS/BODMAS).
First, let's rewrite the expression in a more manageable form:
\[ \frac{9}{8} \div 7 \div 6 \times \frac{14}{3} \times 20 \times \frac{22}{11} \]
Now, let's process the divisions and multiplications step by step:
- Convert divisions to multiplications by taking the reciprocal:
\[ \frac{9}{8} \times \frac{1}{7} \times \frac{1}{6} \times \frac{14}{3} \times 20 \times \frac{22}{11} \]
- Perform multiplication of the fractions:
Now we can multiply the numerators together and the denominators together:
Numerator: \[ 9 \times 1 \times 1 \times 14 \times 20 \times 22 = 9 \times 14 \times 20 \times 22 \]
Denominator: \[ 8 \times 7 \times 6 \times 3 \times 1 \times 11 = 8 \times 7 \times 6 \times 3 \times 11 \]
Calculating the numerator:
- \(9 \times 14 = 126\)
- \(126 \times 20 = 2520\)
- \(2520 \times 22 = 55440\)
Calculating the denominator:
- \(8 \times 7 = 56\)
- \(56 \times 6 = 336\)
- \(336 \times 3 = 1008\)
- \(1008 \times 11 = 11088\)
Now, we have:
\[ \frac{55440}{11088} \]
Next, we simplify this fraction. We can divide both the numerator and the denominator by 11088:
Calculating:
\[ 55440 \div 11088 = 5 \]
So, the expression simplifies to:
\[ \frac{55440}{11088} = 5 \]
Therefore, the final answer is:
\[ \boxed{5} \]