Solution: Converting $315_8$ to decimal: $3 imes 8^2 + 1 imes 8^1 + 5 imes 8^0 = 3 imes 64 + 1 imes 8 + 5 imes 1 = 192 + 8 + 5 = 205$. The base-ten equivalent is $oxed{205}$.
["# Converting $315_8$ to Decimal: The Complete Step-by-Step Solution", "Transforming numbers from base-8 (octal) to base-10 (decimal) is a fundamental skill in number systems, essential for computer science, programming, and digital electronics. If you’ve ever wondered how to convert the octal number $315_8$ into its decimal equivalent, this article walks you through the solution clearly and systematically.", "---", "## What is $315_8$?", "The notation $315_8$ represents a number written in base 8, where each digit corresponds to a power of 8 based on its position from right to left. Let’s break it down.", "### Understanding the positional values in base 8", "In any base system, each digit’s value is determined by its position multiplied by the base raised to that position. For base 8:", "- The rightmost digit is in the $8^0 = 1$ place\n- The middle digit is in the $8^1 = 8$ place\n- The leftmost digit is in the $8^2 = 64$ place", "So the octal number $315_8$ breaks down as:", "| Digit | Position | Value | Place Weight ($8^n$) |\n|-------|----------|-------|-----------------------|\n| 3 | 2 | 3 | $3 \ imes 64 = 192$ |\n| 1 | 1 | 1 | $1 \ imes 8 = 8$ |\n| 5 | 0 | 5 | $5 \ imes 1 = 5$ |", "---", "## Applying the Conversion Formula", "The general formula to convert an octal number to decimal is:\n[\nN_{10} = d_n \ imes 8^n + d_{n-1} \ imes 8^{n-1} + \dots + d_0 \ imes 8^0\n]", "For $315_8$, applying this:\n[\n315_8 = 3 \ imes 8^2 + 1 \ imes 8^1 + 5 \ imes 8^0\n]", "Now compute each term:\n- $3 \ imes 8^2 = 3 \ imes 64 = 192$\n- $1 \ imes 8^1 = 1 \ imes 8 = 8$\n- $5 \ imes 8^0 = 5 \ imes 1 = 5$", "---", "## Final Calculation", "Add the components:\n[\n192 + 8 + 5 = 205\n]", "Thus, the decimal (base-10) equivalent of $315_8$ is\n[\n\boxed{205}\n]", "---", "## Why This Conversion Matters", "Converting octal to decimal is crucial in computing contexts, especially when dealing with low-level system operations, data encoding, and debugging. Understanding this process supports deeper knowledge of numeral systems that underpin modern technology.", "---", "Whether you're a student mastering number systems or a professional working in tech, mastering base conversions like $315_8 \ o 205_{10}$ empowers you with foundational insight into how computers interpret and process numerical data."]
