To solve the problem where \( y \) is inversely proportional to \( x \), we need to understand the concept of inverse proportionality. When we say that \( y \) is inversely proportional to \( x \), it means that as \( x \) increases, \( y \) decreases, and vice versa. Mathematically, this relationship can be expressed as: \[ y = \frac{k}{x} \] where \( k \) is a constant. ### Step 1: Find the constant \( k \) We are given that \( y = 4 \) when \( x = \frac{1}{2} \). We can use this information to find the value of \( k \). Substituting the values into the equation: \[ 4 = \frac{k}{\frac{1}{2}} \] To simplify this, we can multiply both sides by \( \frac{1}{2} \): \[ 4 \cdot \frac{1}{2} = k \] Calculating the left side: \[ 2 = k \] So, we have determined that the constant \( k \) is 2. Now we can rewrite the equation that describes the relationship between \( y \) and \( x \): \[ y = \frac{2}{x} \] ### Step 2: Find \( x \) when \( y = 10 \) Now we need to find the value of \( x \) when \( y = 10 \). We substitute \( y = 10 \) into the equation we derived: \[ 10 = \frac{2}{x} \] To solve for \( x \), we can rearrange the equation. First, multiply both sides by \( x \): \[ 10x = 2 \] Next, divide both sides by 10: \[ x = \frac{2}{10} = \frac{1}{5} \] ### Conclusion Thus, the value of \( x \) when \( y = 10 \) is: \[ \boxed{\frac{1}{5}} \] ### Explanation of Other Options Now, let's analyze the other options to understand why they are incorrect: - **Option A: 2** - If \( x = 2 \), then substituting into the equation \( y = \frac{2}{x} \) gives \( y = \frac{2}{2} = 1 \). This does not equal 10, so this option is incorrect. - **Option B: 10** - If \( x = 10 \), then substituting into the equation gives \( y = \frac{2}{10} = \frac{1}{5} \). This does not equal 10, so this option is also incorrect. - **Option D: 1/10** - If \( x = \frac{1}{10} \), then substituting into the equation gives \( y = \frac{2}{\frac{1}{10}} = 2 \cdot 10 = 20 \). This does not equal 10, so this option is incorrect as well. ### Revision Summary - Inverse proportionality means \( y = \frac{k}{x} \) for some constant \( k \). - To find \( k \), use known values of \( y \) and \( x \). - Rearranging the equation allows you to solve for \( x \) when given a new value of \( y \). - Always check other options to confirm they do not satisfy the equation. This thorough understanding of inverse proportionality and careful calculation will help you tackle similar problems in the future!