In a survey of 120 households, 70 do homestay business and 50 do agriculture business. 30 households do neither. Find how many do both.

Households doing only homestay business:

Households doing only agriculture business:

a) Write the cardinality notation of the number of households who do not do any of the homestay and agriculture business.
The cardinality notation for the number of households who do not do any of the homestay and agriculture business is $n(H \cup A)'$.

b) Present the above information in a Venn-diagram.
The Venn diagram would show:

• A universal set $U$ containing 120 households.
• Two overlapping circles, $H$ (Homestay) and $A$ (Agriculture).
• The intersection $H \cap A$ contains 30 households.
• The part of $H$ only contains $n(H \text{ only}) = 40$ households.
• The part of $A$ only contains $n(A \text{ only}) = 20$ households.
• The region outside both circles, $n(H \cup A)'$, contains 30 households.

       +---------------------------------+
       |             U = 120             |
       |    +-----------------------+    |
       |    |           H           |    |
       |    |     (40)      (30)    |    |
       |    |           +-------+   |    |
       |    |           |       |   |    |
       |    |           |   A   |   |    |
       |    |           | (20)  |   |    |
       |    +-----------+-------+---+    |
       |             (30)              |
       +---------------------------------+

c) Find the number of households doing only one business.
The number of households doing only one business is the sum of households doing only homestay business and households doing only agriculture business.

The number of households doing only one business is $\boxed{60}$.

d) If 10 households who do other work start homestay business, find the ratio of households who do homestay business to agriculture business.
Original number of households doing homestay business, $n(H)_{\text{old}} = 70$.
Original number of households doing agriculture business, $n(A)_{\text{old}} = 50$.
Original number of households doing other work, $n(H \cup A)'_{\text{old}} = 30$.

If 10 households from those doing other work start homestay business:
The new number of households doing homestay business, $n(H)_{\text{new}} = n(H)_{\text{old}} + 10 = 70 + 10 = 80$.
The number of households doing agriculture business remains unchanged, $n(A)_{\text{new}} = 50$.

The ratio of households doing homestay business to agriculture business is:

The ratio of households doing homestay business to agriculture business is $\boxed{8:5}$.