Solution: Given $ \frac{a}{b} = 3 $, let $ a = 3b $. Substituting into $ a + b = 16 $: $ 3b + b = 16 $, so $ b = 4 $ and $ a = 12 $. Now, $ a^2 - b^2 $ factors as $ (a - b)(a + b) = (12 - 4)(16) = 8 \times 16 = 128 $. Therefore, the value is $ \boxed{128} $.

["Title: Step-by-Step Solution to $ \frac{a}{b} = 3 $ and Compute $ a^2 - b^2 $ | A Clear Mathematical Breakdown", "When solving equations involving ratios and algebraic expressions, a structured approach helps uncover the correct result efficiently. In this article, we explore how to compute $ a^2 - b^2 $ given that $ \frac{a}{b} = 3 $ and $ a + b = 16 $, using substitution and factoring. This method combines ratio analysis with algebraic expansion for a precise solution.", "---", "### Step 1: Use the Given Ratio to Set Up an Equation", "We are given:\n$$\n\frac{a}{b} = 3\n$$\nThis means $ a = 3b $.", "---", "### Step 2: Substitute into the Sum Equation", "The second condition is:\n$$\na + b = 16\n$$\nSubstitute $ a = 3b $:\n$$\n3b + b = 16 \quad \Rightarrow \quad 4b = 16\n$$\nSolve for $ b $:\n$$\nb = 4\n$$", "---", "### Step 3: Solve for $ a $", "Using $ a = 3b $:\n$$\na = 3 \ imes 4 = 12\n$$", "---", "### Step 4: Compute $ a^2 - b^2 $ Using Algebraic Factoring", "Recall the difference of squares identity:\n$$\na^2 - b^2 = (a - b)(a + b)\n$$", "We already have:\n- $ a = 12 $, $ b = 4 $\n- $ a + b = 16 $ (given)\n- $ a - b = 12 - 4 = 8 $", "Substitute into the factoring formula:\n$$\na^2 - b^2 = (8)(16) = 128\n$$", "---", "### Final Answer\n$$\n\boxed{128}\n$$", "---", "### Why This Method Works", "Using substitution from the ratio leverages known relationships, while applying the difference of squares factorizes the expression simply. This method combines algebraic reasoning with efficient computation, making it ideal for similar math problems.", "Whether you're solving for variables in equations or evaluating expressions, systematic substitution and key identities (like $ a^2 - b^2 = (a-b)(a+b) $) lead to clear, correct results every time.", "---", "Keywords:\n$ a^2 - b^2 $ factorization, solving equations substitution, difference of squares, $ \frac{a}{b} = 3 $, algebraic expression, step-by-step math solution, mathematical breakdown, solving linear equations.", "---", "### Perfect for Students and Learners\nIf you're studying algebra, mastering substitution and factoring is essential. This example demonstrates how clear reasoning transforms word problems into straightforward calculations. Use it to build confidence in solving similar equations."]