To solve the problem, we need to understand the relationship described in the question. The statement "x varies directly as the product of u and v and inversely as their sum" can be expressed mathematically. Let's break it down step-by-step. ### Step 1: Formulate the Relationship The phrase "x varies directly as the product of u and v" means that: \[ x \propto (u \cdot v) \] This can be expressed as: \[ x = k \cdot (u \cdot v) \] where \( k \) is a constant of proportionality. The phrase "and inversely as their sum" means that: \[ x \propto \frac{1}{(u + v)} \] This can be expressed as: \[ x = \frac{k \cdot (u \cdot v)}{(u + v)} \] Combining these two relationships, we get: \[ x = k \cdot \frac{(u \cdot v)}{(u + v)} \] ### Step 2: Find the Constant \( k \) We are given that \( x = 3 \) when \( u = 3 \) and \( v = 1 \). We can substitute these values into our equation to find \( k \). 1. Calculate \( u \cdot v \): \[ u \cdot v = 3 \cdot 1 = 3 \] 2. Calculate \( u + v \): \[ u + v = 3 + 1 = 4 \] 3. Substitute into the equation: \[ 3 = k \cdot \frac{3}{4} \] 4. Solve for \( k \): \[ k = 3 \cdot \frac{4}{3} = 4 \] ### Step 3: Use \( k \) to Find \( x \) for New Values of \( u \) and \( v \) Now we need to find the value of \( x \) when \( u = 3 \) and \( v = 3 \). 1. Calculate \( u \cdot v \): \[ u \cdot v = 3 \cdot 3 = 9 \] 2. Calculate \( u + v \): \[ u + v = 3 + 3 = 6 \] 3. Substitute these values into the equation: \[ x = 4 \cdot \frac{9}{6} \] 4. Simplify: \[ x = 4 \cdot 1.5 = 6 \] ### Conclusion Thus, the value of \( x \) when \( u = 3 \) and \( v = 3 \) is **6**. ### Explanation of Other Options - **Option A (3)**: This is incorrect because it does not take into account the new values of \( u \) and \( v \) correctly. - **Option B (4)**: This is also incorrect as it does not reflect the calculations based on the new values of \( u \) and \( v \). - **Option D (9)**: This is incorrect because it suggests a higher value than what is calculated based on the relationship defined. ### Revision Summary - **Direct Variation**: Understand that direct variation means the variable increases or decreases proportionally with another variable. - **Inverse Variation**: Recognize that inverse variation means as one variable increases, the other decreases. - **Combining Relationships**: When combining direct and inverse variations, use the formula \( x = k \cdot \frac{(u \cdot v)}{(u + v)} \). - **Finding Constants**: Use given values to find the constant of proportionality, which is crucial for solving for other values. This thorough understanding will help you tackle similar problems in the future!