\(16 = rac{8}{1 - r}\) â \(1 - r = rac{8}{16} = 0.5\) â \(r = 0.5\).
["Solving (16 = \frac{8}{1 - r}) to Find (r = 0.5): A Step-by-Step Guide", "Understanding how to solve simple algebraic equations is essential for mastering basic math and building confidence in algebraic reasoning. One common equation students frequently encounter is:", "[\n16 = \frac{8}{1 - r}\n]", "In this article, we’ll walk through the step-by-step solution to uncover the value of ( r ), demonstrate how simplifications lead to clear results, and explain the practical meaning behind the solution.", "---", "### What Is the Equation (16 = \frac{8}{1 - r})?", "At its core, this equation expresses a relationship between a constant (16) and a fraction involving the unknown variable ( r ). Our goal is to isolate ( r ) and find its numerical value, which turns out to be ( r = 0.5 ).", "---", "### Step-by-Step Solution", "#### Step 1: Eliminate the Fraction\nStart by clearing the denominator. Multiply both sides of the equation by ( 1 - r ):", "[\n16(1 - r) = 8\n]", "This step removes the fraction by multiplying through with the denominator.", "---", "#### Step 2: Distribute and Simplify\nNow distribute the 16 across ( 1 - r ):", "[\n16 - 16r = 8\n]", "This linear expression simplifies the problem by combining constants and isolating the variable term.", "---", "#### Step 3: Isolate the Variable Term\nSubtract 16 from both sides:", "[\n-16r = 8 - 16\n]", "[\n-16r = -8\n]", "Now, both sides are simpler, making it easier to solve for ( r ).", "---", "#### Step 4: Solve for ( r )\nDivide both sides by (-16):", "[\nr = \frac{-8}{-16} = \frac{1}{2} = 0.5\n]", "Thus, we find:", "[\nr = 0.5\n]", "---", "### Why Is This Simplification (1 - r = \frac{8}{16}) Important?", "Returning to the original manipulation, when we divided both sides by 16, we obtained:", "[\n1 - r = \frac{8}{16} = 0.5\n]", "This reveals a clean connection: the denominator ( 1 - r ) equals ( \frac{8}{16} ), which directly yields the value of ( r ) when rearranged. It also confirms the equation’s symmetry and logical consistency — a valuable concept in algebra for checking work.", "---", "### Final Answer:\n[\nr = 0.5\n]", "---", "### Practical Use of This Result", "Understanding how to solve equations like (16 = \frac{8}{1 - r}) prepares learners for more complex problems in finance, physics, and data modeling, where such relationships model growth, decay, interest rates, or proportional reasoning.", "---", "### Summary", "- Begin with (16 = \frac{8}{1 - r})\n- Multiply both sides by (1 - r)\n- Simplify using distribution and arithmetic\n- Isolate (r) and solve\n- Verify the result through equivalent forms like (1 - r = 0.5)\n- Confirm (r = 0.5)", "This clear, systematic approach ensures accuracy and builds foundational algebra skills.", "---", "Keywords:\nfrACTION solves, 16 equals 8 over (1 minus r) solved, 16 = 8⁄(1 - r) solution, algebra equation r = 0.5, step-by-step linear equation, solving for r, algebraic identities, fractional equations, simple algebra practice, r = 0.5 meaning."]
