In 2020, the ratio of the number of men to the number of women in Happy Factory was 2 : 1. The ratio of the number of men to the number of women in Cheery Factory was 7 : 13. The total number of people in Happy Factory was 90% of the total number of people in Cheery Factory.

1. Express the number of men and women in Cheery Factory in 2020 using units

From the given ratios and the $90\%$ condition (as found in part a):
Total people in Cheery Factory $= 7 + 13 = 20$ units.
Number of men in Cheery Factory $= 7$ units.
Number of women in Cheery Factory $= 13$ units.

2. Set up an equation using the new ratio in 2021

When $8$ men join Cheery Factory, the number of men becomes $7\text{ units} + 8$, while the number of women remains $13\text{ units}$.
The new ratio of men to women is $3 : 5$.
$\frac{7\text{ units} + 8}{13\text{ units}} = \frac{3}{5}$

3. Solve for the value of 1 unit

$5(7\text{ units} + 8) = 3(13\text{ units})$
$35\text{ units} + 40 = 39\text{ units}$
$4\text{ units} = 40$
$1\text{ unit} = 10$

4. Calculate the total number of people in Cheery Factory in 2020

Total number of people in Cheery Factory in 2020 $= 20\text{ units}$
$20 \times 10 = 200$