To solve the equation \(2_9 \times (Y3)_9 = 3_5 \times (Y3)_5\), we need to first understand what the numbers in different bases mean and how to convert them to a common base (usually base 10) for easier calculations. ### Step 1: Convert the numbers from their respective bases to base 10. 1. **Convert \(2_9\) to base 10:** - In base 9, \(2_9\) is simply \(2\) in base 10. 2. **Convert \((Y3)_9\) to base 10:** - The number \((Y3)_9\) can be expressed as: \[ Y \times 9^1 + 3 \times 9^0 = 9Y + 3 \] 3. **Convert \(3_5\) to base 10:** - In base 5, \(3_5\) is simply \(3\) in base 10. 4. **Convert \((Y3)_5\) to base 10:** - The number \((Y3)_5\) can be expressed as: \[ Y \times 5^1 + 3 \times 5^0 = 5Y + 3 \] ### Step 2: Substitute the base 10 values into the equation. Now we can rewrite the original equation using the base 10 conversions: \[ 2 \times (9Y + 3) = 3 \times (5Y + 3) \] ### Step 3: Expand both sides of the equation. 1. **Left side:** \[ 2 \times (9Y + 3) = 18Y + 6 \] 2. **Right side:** \[ 3 \times (5Y + 3) = 15Y + 9 \] ### Step 4: Set the two sides equal to each other. Now we have: \[ 18Y + 6 = 15Y + 9 \] ### Step 5: Solve for \(Y\). 1. Subtract \(15Y\) from both sides: \[ 18Y - 15Y + 6 = 9 \] This simplifies to: \[ 3Y + 6 = 9 \] 2. Subtract \(6\) from both sides: \[ 3Y = 3 \] 3. Divide both sides by \(3\): \[ Y = 1 \] ### Conclusion: The value of \(Y\) is \(1\). Therefore, the correct option is **D. 1**. ### Explanation of Other Options: - **Option A (4):** If \(Y = 4\), substituting back into the equation would not satisfy the equality, as the left side would be larger than the right side. - **Option B (3):** If \(Y = 3\), substituting back would also not satisfy the equality, as the left side would again be larger than the right side. - **Option C (2):** If \(Y = 2\), substituting back would yield a left side that is still larger than the right side, thus not satisfying the equation. ### Revision Summary: - Convert numbers from their respective bases to base 10 for easier calculations. - Set the equation after conversion and simplify it step by step. - Solve for the variable by isolating it on one side of the equation. - Check each option by substituting back into the original equation to verify correctness. This methodical approach ensures clarity and accuracy in solving base conversion problems in mathematics.