Divide both sides by $ rac{16}{3} $:

["# How to Divide Both Sides by $ \frac{16}{3} $: A Step-by-Step Guide", "Understanding how to divide both sides of an equation by a fraction is a fundamental algebra skill that helps solve equations efficiently. One common problem you may encounter is dividing both sides of an equation by $ \frac{16}{3} $. In this SEO-optimized article, we’ll explain the process clearly, provide examples, and highlight key tips to master this operation.", "---", "## What Does “Divide Both Sides by $ \frac{16}{3} $” Mean?", "Dividing both sides of an equation by a number (or fraction) means reducing the equation’s balance while preserving equality. If you divide both sides by $ \frac{16}{3} $, you effectively multiply each side by the reciprocal: $ \frac{3}{16} $. This operation simplifies expressions involving division by fractions, making it easier to isolate variables.", "---", "## Step-by-Step: Dividing Both Sides by $ \frac{16}{3} $", "### Step 1: Write the Original Equation\nStart with an equation involving $ \frac{16}{3} $, such as:\n$$\n\frac{3x + 6}{16/3} = 9\n$$", "### Step 2: Simplify Division by a Fraction\nRemember: dividing by $ \frac{16}{3} $ is the same as multiplying by $ \frac{3}{16} $. Rewrite the equation:\n$$\n\left( \frac{3x + 6}{16/3} \right) \ imes \frac{3}{16} = 9\n$$\nor equivalently:\n$$\n\left( \frac{3x + 6}{16} \right) \ imes 3 = 9\n$$", "### Step 3: Multiply Both Sides by $ \frac{3}{16} $\nMultiply each side to eliminate the denominator:\n$$\n3x + 6 = 9 \ imes \frac{16}{3}\n$$", "### Step 4: Simplify the Right Side\nCompute $ 9 \div \frac{16}{3} = 9 \ imes \frac{3}{16} = \frac{27}{16} $? Wait — actually, since we multiplied by $ \frac{3}{16} $, the result is:\n$$\n9 \ imes \frac{3}{16} = \frac{27}{16}\n$$\nBut caution: actually dividing by $ \frac{16}{3} $ means multiplying by $ \frac{3}{16} $, so:\n$$\n9 \ imes \frac{3}{16} = \frac{27}{16}\n$$\nWait — correction: the division step is:\n$$\n\frac{3x + 6}{16/3} = (3x + 6) \cdot \frac{3}{16}\n$$\nThen after dividing both sides by $ \frac{16}{3} $, you multiply both sides by $ \frac{3}{16} $:\n$$\n3x + 6 = 9 \cdot \frac{3}{16} = \frac{27}{16}\n$$", "But better:\nFrom:\n$$\n\frac{3x + 6}{16/3} = 9\n\Rightarrow (3x + 6) \cdot \frac{3}{16} = 9\n\Rightarrow 3x + 6 = 9 \cdot \frac{16}{3} = 48\n$$\nYes — because $ \frac{a}{b/c} = a \cdot \frac{c}{b} $, so $ \frac{3x + 6}{16/3} = (3x + 6) \cdot \frac{3}{16} $, then multiplying both sides by $ \frac{3}{16} $ gives:\n$$\n3x + 6 = 9 \cdot \frac{16}{3} \cdot \frac{3}{16} \quad \ ext{No — misunderstanding.}\n$$", "Let’s clarify:", "Correct approach:\nTo divide both sides by $ \frac{16}{3} $, multiply both sides by its reciprocal $ \frac{3}{16} $:\n$$\n\frac{3x + 6}{16/3} \div \frac{16}{3} = 9 \quad \Rightarrow \quad \frac{3x + 6}{16/3} \cdot \frac{3}{16} = 9\n$$\nNow simplify the product:\n$$\n\frac{3x + 6}{16/3} \cdot \frac{3}{16} = (3x + 6) \cdot \frac{3}{16} \cdot \frac{3}{16} = (3x + 6) \cdot \frac{9}{256}\n$$\nThat’s messy — better: instead, recognizing that dividing by $ \frac{16}{3} $ is same as multiplying by $ \frac{3}{16} $, so:", "Start over clearly:", "---", "### Simplified Correct Process:", "Given:\n$$\n\frac{3x + 6}{16/3} = 9\n$$", "Step 1: Multiply both sides by $ \frac{3}{16} $:\n$$\n\frac{3x + 6}{16/3} \ imes \frac{3}{16} = 9 \ imes \frac{3}{16}\n$$\nBut more cleanly: dividing by $ \frac{16}{3} $ is multiply by $ \frac{3}{16} $, so:\n$$\n(3x + 6) \cdot \frac{3}{16} = 9 \cdot \frac{3}{16}\n$$", "Wait — no: $ \frac{A}{B} \div B = A \cdot \frac{1}{B} = A \cdot \frac{1}{B} $, and $ \frac{1}{B} = \frac{\ ext{reciprocal}}{1} $, so $ \frac{A}{B} \cdot \frac{1}{B} = \frac{A}{B^2} $? No — $ \frac{A}{B} \ imes \frac{1}{B} = \frac{A}{B^2} $. That’s correct.", "But better: since dividing by $ \frac{16}{3} $ is same as multiplying by $ \frac{3}{16} $, we write:\n$$\n(3x + 6) \ imes \frac{3}{16} = 9 \ imes \frac{3}{16}\n$$", "Now factor $ \frac{3}{16} $ out:\n$$\n\frac{3}{16} (3x + 6) = \frac{27}{16}\n$$", "Now divide both sides by $ \frac{3}{16} $? No — we just simplified:\n$$\n(3x + 6) \cdot \frac{3}{16} = \frac{27}{16}\n$$\nSo divide both sides by $ \frac{3}{16} $:\n$$\n3x + 6 = \frac{27/16}{3/16} = \frac{27}{16} \cdot \frac{16}{3} = 9\n$$", "Alternatively, simply:\n$$\n(3x + 6) \cdot \frac{3}{16} = \frac{27}{16} \Rightarrow 3x + 6 = 9\n$$", "Step 4: Solve for $ x $:\n$$\n3x + 6 = 9\n\Rightarrow 3x = 3\n\Rightarrow x = 1\n$$", "---", "## Final Answer", "Dividing both sides of the equation by $ \frac{16}{3} $ yields:\n$$\n3x + 6 = 9 \quad \Rightarrow \quad x = 1\n$$", "---", "## Why This Works: Key Takeaways", "- Dividing by a fraction equals multiplying by its reciprocal.\n- Simplifying expressions carefully avoids errors.\n- Maintaining balance is essential in algebra.\n- Practice transforms abstract steps into intuitive problem-solving.", "---", "## FAQ: Common Questions About Dividing by Fractions", "Q: Why do we multiply by the reciprocal instead of dividing?\nA: Division by $ \frac{a}{b} $ means multiplying by $ \frac{b}{a} $ to preserve the meaning of division.", "Q: Can I just divide by $ \frac{16}{3} $ directly without reciprocal?\nA: Yes and no — you can rewrite division as multiplication, but using the reciprocal simplifies arithmetic.", "Q: What if the denominator is a fraction?\nA: Learning to divide by fractions is essential — just multiply by the reciprocal.", "---", "## Practice Problems", "Try dividing both sides of the following by $ \frac{16}{3} $:\n1. $ \frac{2y - 4}{16/3} = 12 $\n2. $ \frac{x + 5}{8} \div \frac{16}{3} = 3 $", "Solutions:\n1. Multiply both sides by $ \frac{3}{16} $: $ \frac{2y - 4}{16/3} = 12 \Rightarrow (2y - 4) \cdot \frac{3}{16} = 12 \Rightarrow 2y - 4 = 12 \cdot \frac{16}{3} = 64 \Rightarrow 2y = 68 \Rightarrow y = 34 $", "2. $ \frac{x + 5}{8} \cdot \frac{3}{16} = 3 \Rightarrow \frac{3(x + 5)}{128} = 3 \Rightarrow 3(x + 5) = 384 \Rightarrow x + 5 = 128 \Rightarrow x = 123 $", "---", "## Conclusion", "Dividing both sides by $ \frac{16}{3} $ is a standard algebraic technique that reinforces understanding of fractions, reciprocals, and equation solving. By mastering this step, students build confidence in manipulating equations and prepare for more complex algebra. Use this guide to clarify your steps, double-check your work, and solve with clarity.", "---", "Keywords:\ndivide both sides by 16/3, algebra equation solving, fraction division, multiplying by reciprocal, solving linear equations, step-by-step division, how to divide by a fraction, algebraic operations, math practice", "Meta Description:\nLearn how to divide both sides of an equation by $ \frac{16}{3} $ with step-by-step instructions, examples, and key algebra concepts to improve your equation-solving skills. Perfect for students mastering fractions and linear equations."]