To solve the equation \( 451_6 - P_7 = 305_6 \), we need to convert all numbers from their respective bases (base 6 and base 7) into base 10 (decimal) so that we can perform the arithmetic operations easily. ### Step 1: Convert \( 451_6 \) to Decimal To convert a number from base 6 to decimal, we use the formula: \[ \text{Decimal} = d_n \cdot b^n + d_{n-1} \cdot b^{n-1} + \ldots + d_1 \cdot b^1 + d_0 \cdot b^0 \] where \( d \) are the digits of the number and \( b \) is the base. For \( 451_6 \): - The digit \( 4 \) is in the \( 6^2 \) place. - The digit \( 5 \) is in the \( 6^1 \) place. - The digit \( 1 \) is in the \( 6^0 \) place. Calculating this gives: \[ 451_6 = 4 \cdot 6^2 + 5 \cdot 6^1 + 1 \cdot 6^0 \] \[ = 4 \cdot 36 + 5 \cdot 6 + 1 \cdot 1 \] \[ = 144 + 30 + 1 = 175 \] So, \( 451_6 = 175_{10} \). ### Step 2: Convert \( 305_6 \) to Decimal Now, we convert \( 305_6 \) to decimal using the same method: For \( 305_6 \): - The digit \( 3 \) is in the \( 6^2 \) place. - The digit \( 0 \) is in the \( 6^1 \) place. - The digit \( 5 \) is in the \( 6^0 \) place. Calculating this gives: \[ 305_6 = 3 \cdot 6^2 + 0 \cdot 6^1 + 5 \cdot 6^0 \] \[ = 3 \cdot 36 + 0 \cdot 6 + 5 \cdot 1 \] \[ = 108 + 0 + 5 = 113 \] So, \( 305_6 = 113_{10} \). ### Step 3: Set Up the Equation Now we can rewrite the original equation in decimal: \[ 175 - P_7 = 113 \] ### Step 4: Solve for \( P_7 \) Rearranging the equation to solve for \( P_7 \): \[ P_7 = 175 - 113 \] \[ P_7 = 62 \] ### Step 5: Convert \( 62_{10} \) to Base 7 Now we need to convert \( 62 \) from decimal to base 7. We do this by dividing the number by 7 and keeping track of the remainders. 1. \( 62 \div 7 = 8 \) remainder \( 6 \) 2. \( 8 \div 7 = 1 \) remainder \( 1 \) 3. \( 1 \div 7 = 0 \) remainder \( 1 \) Reading the remainders from bottom to top, we find that \( 62_{10} = 116_7 \). ### Step 6: Find the Final Answer Since \( P_7 = 62_{10} \) corresponds to \( 116_7 \), we need to express \( P \) in decimal. The options provided are in decimal, so we need to convert \( 116_7 \) back to decimal to find the correct option. Calculating \( 116_7 \): - The digit \( 1 \) is in the \( 7^2 \) place. - The digit \( 1 \) is in the \( 7^1 \) place. - The digit \( 6 \) is in the \( 7^0 \) place. Calculating this gives: \[ 116_7 = 1 \cdot 7^2 + 1 \cdot 7^1 + 6 \cdot 7^0 \] \[ = 1 \cdot 49 + 1 \cdot 7 + 6 \cdot 1 \] \[ = 49 + 7 + 6 = 62 \] ### Conclusion The value of \( P \) in decimal is \( 62 \). However, the options provided do not include \( 62 \). Let's check the options again: - A. 627 - B. 1167 - C. 6117 - D. 1427 None of these options match \( 62 \). It seems there was a misunderstanding in the options provided. ### Summary - Convert numbers from their respective bases to decimal for easier calculations. - Perform arithmetic operations in decimal. - Convert the result back to the required base if necessary. - Always double-check the options provided to ensure they match the calculated results. **Revision Summary:** - Convert base numbers to decimal for calculations. - Rearrange equations to isolate variables. - Convert results back to the required base if needed. - Verify options against calculated results.