A survey of 80 farmers found that 50 grow maize and 60 grow rice. Represent this information on a Venn diagram and find the number of farmers who grow both crops.

Here's how to solve the problem:

a) Represent the information on a Venn diagram.
Let $M$ be the set of farmers who grow maize and $R$ be the set of farmers who grow rice.
Given:
Total farmers, $n(U) = 80$
Farmers who grow maize, $n(M) = 50$
Farmers who grow rice, $n(R) = 60$
Let $x$ be the number of farmers who grow both crops, $n(M \cap R) = x$.

The regions in the Venn diagram are:

• Farmers who grow only maize: $n(M \text{ only}) = n(M) - n(M \cap R) = 50 - x$
• Farmers who grow only rice: $n(R \text{ only}) = n(R) - n(M \cap R) = 60 - x$
• Farmers who grow both maize and rice: $n(M \cap R) = x$

The Venn diagram would show two overlapping circles. The left part of the maize circle (not overlapping) would be $50-x$. The overlapping part would be $x$. The right part of the rice circle (not overlapping) would be $60-x$.

b) Write an equation in $x$ and solve it.
The total number of farmers is the sum of farmers in each distinct region of the Venn diagram.

$$n(U) = n(M \text{ only}) + n(R \text{ only}) + n(M \cap R)$$

Substitute the expressions for each region:

$$80 = (50 - x) + (60 - x) + x$$

Combine like terms:

$$80 = 50 + 60 - x - x + x$$ $$80 = 110 - x$$

Now, solve for $x$:

$$x = 110 - 80$$ $$x = 30$$

The number of farmers who grow both crops is $\boxed{\text{30}}$.