To solve the problem of finding the average of four numbers given their sum in base five, we will follow a systematic approach. Let's break it down step-by-step. ### Step 1: Understanding the Sum in Base Five The sum of the four numbers is given as \(1214_5\). This means that the number is in base five, and we need to convert it to base ten (decimal) to perform our calculations. **Conversion from Base Five to Base Ten:** The base five number \(1214_5\) can be converted to decimal using the formula: \[ n = d_k \cdot b^k + d_{k-1} \cdot b^{k-1} + \ldots + d_1 \cdot b^1 + d_0 \cdot b^0 \] Where \(d\) are the digits, \(b\) is the base (which is 5 in this case), and \(k\) is the position of the digit from the right (starting from 0). For \(1214_5\): - The digit \(1\) is in the \(3^{rd}\) position (from the right), so it contributes \(1 \cdot 5^3\). - The digit \(2\) is in the \(2^{nd}\) position, contributing \(2 \cdot 5^2\). - The digit \(1\) is in the \(1^{st}\) position, contributing \(1 \cdot 5^1\). - The digit \(4\) is in the \(0^{th}\) position, contributing \(4 \cdot 5^0\). Calculating each term: - \(1 \cdot 5^3 = 1 \cdot 125 = 125\) - \(2 \cdot 5^2 = 2 \cdot 25 = 50\) - \(1 \cdot 5^1 = 1 \cdot 5 = 5\) - \(4 \cdot 5^0 = 4 \cdot 1 = 4\) Now, we sum these contributions: \[ 125 + 50 + 5 + 4 = 184 \] So, \(1214_5\) in decimal is \(184\). ### Step 2: Finding the Average To find the average of the four numbers, we divide the total sum by the number of numbers (which is 4): \[ \text{Average} = \frac{\text{Sum}}{\text{Number of Values}} = \frac{184}{4} = 46 \] ### Step 3: Converting the Average Back to Base Five Now, we need to convert the average \(46\) from decimal back to base five. **Conversion from Decimal to Base Five:** To convert a decimal number to base five, we repeatedly divide the number by 5 and keep track of the remainders. 1. \(46 \div 5 = 9\) with a remainder of \(1\). 2. \(9 \div 5 = 1\) with a remainder of \(4\). 3. \(1 \div 5 = 0\) with a remainder of \(1\). Now, we read the remainders from bottom to top, which gives us \(141_5\). ### Conclusion: The Correct Option The average expressed in base five is \(141_5\), which corresponds to option **B**. ### Explanation of Other Options - **Option A (114)**: This would represent a smaller average than calculated. In decimal, \(114_5\) converts to \(29\) in decimal, which is not correct. - **Option C (401)**: This represents a larger average. In decimal, \(401_5\) converts to \(101\), which is also incorrect. - **Option D (411)**: This is another incorrect option. In decimal, \(411_5\) converts to \(102\), which does not match our calculated average. ### Revision Summary - Convert the base five number to decimal to find the sum. - Calculate the average by dividing the sum by the number of values. - Convert the average back to base five for the final answer. - The correct average in base five is \(141_5\) (Option B).