Substituting the values, \( S = \frac{5}{1 - \frac{1}{3}} = \frac{5}{\frac{2}{3}} = 5 \times \frac{3}{2} = 7.5 \).
["Understanding Substitution in Mathematics: Simplifying ( S = \frac{5}{1 - \frac{1}{3}} ) Step-by-Step", "In mathematics, substitution is a powerful technique that simplifies complex expressions into manageable forms. One classic example involves cracking the value of ( S ) defined as:", "[\nS = \frac{5}{1 - \frac{1}{3}} = \frac{5}{\frac{2}{3}} = 5 \ imes \frac{3}{2} = 7.5\n]", "In this article, we explore how substitution transforms this expression step by step, clarifying each operation and demonstrating why the result is ( S = 7.5 ).", "---", "### Step 1: Substitute the Denominator Expression", "The expression begins with a fraction:", "[\nS = \frac{5}{1 - \frac{1}{3}}\n]", "The key to solving this lies in evaluating the denominator first. Substitution begins by simplifying the denominator:", "[\n1 - \frac{1}{3}\n]", "Since multiplication by ( 1 ) does not change the value, rewrite ( 1 ) as ( \frac{3}{3} ):", "[\n1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}\n]", "This substitution replaces the entire denominator with its simplified fractional form:", "[\nS = \frac{5}{\frac{2}{3}}\n]", "---", "### Step 2: Division by a Fraction Equals Multiplication by Its Reciprocal", "A fundamental rule in arithmetic states that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore:", "[\n\frac{5}{\frac{2}{3}} = 5 \ imes \frac{3}{2}\n]", "This substitution turns a division problem into a multiplication one, making computation straightforward.", "---", "### Step 3: Multiply to Simplify the Expression", "Now multiply the numerator ( 5 ) by ( \frac{3}{2} ):", "[\n5 \ imes \frac{3}{2} = \frac{5 \ imes 3}{2} = \frac{15}{2}\n]", "Converting to decimal form gives:", "[\n\frac{15}{2} = 7.5\n]", "Thus, the final value of ( S ) is:", "[\nS = 7.5\n]", "---", "### Why Substitution Matters", "This example showcases how substitution—replacing complex segments with simplified equivalents—eases computation. By systematically:", "- Evaluating inner expressions ((1 - \frac{1}{3} = \frac{2}{3})),\n- Converting division to multiplication via reciprocal rules, and\n- Performing clear multiplication steps,", "we transform a potentially confusing fraction into a straightforward calculation.", "---", "### Takeaways", "- Always simplify fractions inside denominators before substituting.\n- Remember that dividing by a fraction means multiplying by its reciprocal.\n- Substitution is not just symbolic—it’s a strategy to reduce complexity.", "Understanding this process helps students confidently tackle algebraic fractions and improves proficiency in mathematical manipulation.", "---", "Key Takeaway:\nSubstituting and simplifying expressions step-by-step—such as calculating\n[\nS = \frac{5}{1 - \frac{1}{3}} = \frac{5}{\frac{2}{3}} = 5 \ imes \frac{3}{2} = 7.5\n]\n– is essential for mastering fraction arithmetic and preparing for advanced math topics.", "---", "SEO keywords for this article:\nFraction simplification, substitution in algebra, dividing fractions, computational math, solving equations step-by-step, mathematical operations explained, value of ( S ), algebra tutorial, simplifying ( S = \frac{5}{1 - \frac{1}{3}} )"]
