This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the solution to the problem. Let M be the set of farmers who grow maize and R be the set of farmers who grow rice. Given: Total number of farmers, |M R| = 80 Number of farmers who grow maize, |M| = 50 Number of farmers who grow rice, |R| = 60 a) Represent the information on a Venn diagram. Let x be the number of farmers who grow both maize and rice, i.e., |M R| = x. Using the Principle of Inclusion-Exclusion: |M R| = |M| + |R| - |M R| 80 = 50 + 60 - x 80 = 110 - x x = 110 - 80 x = 30 So, 30 farmers grow both crops. Number of farmers who grow only maize: |M| - x = 50 - 30 = 20. Number of farmers who grow only rice: |R| - x = 60 - 30 = 30. A Venn diagram would show two overlapping circles. • The intersection (farmers growing both maize and rice) contains 30 farmers. • The part of the maize circle outside the intersection (farmers growing only maize) contains 20 farmers. • The part of the rice circle outside the intersection (farmers growing only rice) contains 30 farmers. The sum of these parts is 20 + 30 + 30 = 80, which is the total number of farmers. b) If x of them grow both crops, write an equation in x and solve for it. Step 1: Use the Principle of Inclusion-Exclusion. |M R| = |M| + |R| - |M R| Step 2: Substitute the given values into the equation. 80 = 50 + 60 - x Step 3: Simplify the equation. 80 = 110 - x Step 4: Solve for x. x = 110 - 80 x = 30 The value of x is 30. c) Find the number of farmers who grow: i. only rice. This is the number of farmers in set R but not in set M. |R only| = |R| - |M R| |R only| = 60 - 30 |R only| = 30 The number of farmers who grow only rice is 30. ii. only maize. This is the number of farmers in set M but not in set R. |M only| = |M| - |M R| |M only| = 50 - 30 |M only| = 20 The number of farmers who grow only maize is 20. iii. either rice or maize. This refers to the total number of farmers who grow at least one of the crops, which is given as the total number of farmers in the village. |M R| = 80 The number of farmers who grow either rice or maize is 80. What's next?