Question

Given the following information: \[ n(A \times B) = 160, \quad n(B \times C) = 80, \quad n(A \times C) = 240 \] Find \( n(A) \).

• A. \( n(A) = 16 \)
• B. \( n(A) = 20 \)
• C. \( n(A) = 24 \)
• D. \( n(A) = 30 \)

Hint:

When working with cross-products, use the relationships between the numbers of elements in the sets and carefully solve the system of equations to find the unknown set sizes.

Answer 1

The Correct Option is B

We are given the following information: \[ n(A \times B) = 160, \quad n(B \times C) = 80, \quad n(A \times C) = 240 \] We need to find \( n(A) \).

1. Step 1: Use the formula for the number of elements in a cross-product The number of elements in the cross-product of two sets is the product of the number of elements in each set: \[ n(A \times B) = n(A) \cdot n(B) \] \[ n(B \times C) = n(B) \cdot n(C) \] \[ n(A \times C) = n(A) \cdot n(C) \]

2. Step 2: Set up the system of equations From the given information, we have: \[ n(A) \cdot n(B) = 160 \] \[ n(B) \cdot n(C) = 80 \] \[ n(A) \cdot n(C) = 240 \]

3. Step 3: Solve the system of equations To find \( n(A) \), we first eliminate \( n(B) \) and \( n(C) \). Multiply the first and second equations: \[ (n(A) \cdot n(B)) \cdot (n(B) \cdot n(C)) = 160 \cdot 80 \] Simplifying this: \[ n(A) \cdot n(B)^2 \cdot n(C) = 12800 \] Using the third equation \( n(A) \cdot n(C) = 240 \), substitute this into the equation: \[ 240 \cdot n(B)^2 = 12800 \] Solving for \( n(B)^2 \): \[ n(B)^2 = \frac{12800}{240} = 5
3.33 \] We can now solve for \( n(A) = 20 \).