Yaseen was travelling from City A to City B. He travelled the first one-fifth of the distance at 10 km/h, and the next three-twentieth of the distance at 15 km/h. Of the remaining distance, he travelled one part at 20 km/h and the rest of the distance at 12 km/h. If the average speed of Yaseen for the entire trip was 16 km/h, what part of the entire trip did Yaseen cover at 20 km/h?

Let the total distance from City A to City B be $D$ km.

First part: $\frac{1}{5}D$ at 10 km/h

Second part: $\frac{3}{20}D$ at 15 km/h

Distance covered in first two parts:

$\frac{1}{5}D + \frac{3}{20}D$

$= \frac{4}{20}D + \frac{3}{20}D$

$= \frac{7}{20}D$

Remaining distance:

$D - \frac{7}{20}D$

$= \frac{13}{20}D$

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Let Yaseen cover a fraction $x$ of the total distance at 20 km/h.

Then the rest at 12 km/h would be: $\frac{13}{20}D - xD$

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Time for first part: $\frac{\frac{1}{5}D}{10}$

$= \frac{D}{50}$

Time for second part: $\frac{\frac{3}{20}D}{15}$

$= \frac{3D}{300}$

$= \frac{D}{100}$

Time for third part (at 20 km/h): $\frac{xD}{20}$

Time for fourth part (at 12 km/h): $\frac{\frac{13}{20}D - xD}{12}$

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Using the average speed formula:

$\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$

$16 = \frac{D}{\frac{D}{50} + \frac{D}{100} + \frac{xD}{20} + \frac{\frac{13}{20}D - xD}{12}}$

Dividing both sides by $D$:

$\frac{1}{\frac{1}{50} + \frac{1}{100} + \frac{x}{20} + \frac{\frac{13}{20} - x}{12}} = 16$

$\frac{1}{50} + \frac{1}{100} + \frac{x}{20} + \frac{\frac{13}{20} - x}{12} = \frac{1}{16}$

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Simplifying the left side:

$\frac{1}{50} + \frac{1}{100} = \frac{2}{100} + \frac{1}{100} = \frac{3}{100}$

$\frac{x}{20} + \frac{\frac{13}{20} - x}{12} = \frac{x}{20} + \frac{13}{240} - \frac{x}{12}$

$= x\left(\frac{1}{20} - \frac{1}{12}\right) + \frac{13}{240}$

$= x\left(\frac{3}{60} - \frac{5}{60}\right) + \frac{13}{240}$

$= -\frac{x}{30} + \frac{13}{240}$

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Substituting back:

$\frac{3}{100} - \frac{x}{30} + \frac{13}{240} = \frac{1}{16}$

Converting to common denominator (1200):

$\frac{36}{1200} - \frac{40x}{1200} + \frac{65}{1200} = \frac{75}{1200}$

$36 - 40x + 65 = 75$

$101 - 40x = 75$

$40x = 26$

$x = \frac{26}{40}$

$x = \frac{13}{20}$

Therefore, Yaseen covered $\frac{13}{20}$ of the entire trip at 20 km/h.