P_{\text{new}} = 6 \times (r + 2) = 6 \times (10 + 2) = 72 \text{ cm}

["## Understanding the Calculation: ( P_{\ ext{new}} = 6 \ imes (r + 2) = 6 \ imes (10 + 2) = 72 , \ ext{cm} )", "When faced with a mathematical expression like ( P_{\ ext{new}} = 6 \ imes (r + 2) ), simplifying and solving it step-by-step is essential—especially when applying real-world measurements such as ( P_{\ ext{new}} = 72 , \ ext{cm} ). In this article, we explain how substitution and basic algebra yield the final value, clarify key terms like ( r ), and emphasize the importance of understanding unit consistency in such calculations.", "---", "### Breaking Down the Formula", "The equation ( P_{\ ext{new}} = 6 \ imes (r + 2) = 6 \ imes (10 + 2) = 72 , \ ext{cm} ) involves a proportional relationship where a constant multiplier (6) scales an adjusted variable ( r + 2 ). Let’s unpack each component:", "- ( P_{\ ext{new}} ): This represents the updated measurement—specifically, a length given in centimeters (cm).\n- ( r ): This is the unknown variable representing a base or reference value (units assumed to be centimeters based on context).\n- ( (r + 2) ): This expression combines the base value ( r ) with an additive adjustment of 2 cm.\n- The brackets ( ( ) ) ensure proper order of operations, indicating that ( r + 2 ) is fully evaluated before multiplication.", "---", "### Step-by-Step Simplification", "Substitute ( r + 2 = 10 + 2 ) into the formula:", "1. Start with the original expression:\n [\n P_{\ ext{new}} = 6 \ imes (r + 2)\n ]", "2. Apply the known value ( r + 2 = 12 ) (since ( 10 + 2 = 12 )):\n [\n P_{\ ext{new}} = 6 \ imes 12\n ]", "3. Multiply to find the final result:\n [\n P_{\ ext{new}} = 72 , \ ext{cm}\n ]", "This computation confirms how a simple linear expression reduces to a real-world measurement when substituting the base variable.", "---", "### Why It Matters: Context and Applications", "Such calculations commonly appear in geometry, physics, and engineering when adjusting dimensions. For instance, calculating a new perimeter, scaled area, or adjusted side length often relies on modified formulas—like multiplying a base value by a factor and adjusting by known dimensions.", "When ( P_{\ ext{new}} = 72 , \ ext{cm} ), this indicates that after applying the formula with ( r = 10 , \ ext{cm} ), the updated measurement doubles the original base length (via ( r + 2 = 12 )) and scales it evenly by 6. This kind of transformation is foundational for design scaling, construction measurements, and educational problem-solving.", "---", "### Key Takeaways", "- Always evaluate parentheses before multiplication to avoid errors.\n- Clarify units — all values here are in centimeters—ensuring meaningful interpretation.\n- Substitution is powerful: plugging known values simplifies complex expressions into concrete numbers.\n- Contextual understanding helps translate mathematical steps into practical applications.", "---", "### Final Thoughts", "The equation ( P_{\ ext{new}} = 6 \ imes (r + 2) ) simplified to 72 cm serves as a clear example of proportional reasoning with real measurement units. By following algebraic steps carefully and interpreting variables in context, even complex formulas become accessible. Whether in school lessons or professional practice, mastering such calculations strengthens logical thinking and problem-solving.", "For faster reference:\n[\nP_{\ ext{new}} = 6(r + 2),\quad r + 2 = 12,\quad P_{\ ext{new}} = 72 , \ ext{cm}\n]", "---", "Keywords: ( P_{\ ext{new}} = 6(r + 2) ), mathematical simplification, unit conversion, algebra in real-world measurement, solving linear equations, proportional reasoning.\nMeta Description: Learn how to simplify ( P_{\ ext{new}} = 6(r + 2) ) step-by-step, apply ( r = 10 ), and verify ( P_{\ ext{new}} = 72 , \ ext{cm} )—a clear guide for math students and professionals."]