Question:** The average of \( 3v+2 \), \( 5v-1 \), and \( 4v+7 \) is required. If \( v \) is a positive integer and the average must be less than 25, what is the maximum possible value of \( v \)?
["Optimize with Confidence: Finding the Maximum Integer Value of ( v ) for a Conditional Average", "When solving math problems involving averages, especially with variables, clarity and precision are key—especially when constraints like integer values and thresholds are involved. In this article, we’ll break down the problem step by step: calculating the average of three expressions, applying the condition that the average must be less than 25, and determining the maximum positive integer value of ( v ) that satisfies this requirement.", "---", "### Step 1: Understand the Average of Three Expressions", "We are given three expressions:", "- ( 3v + 2 )\n- ( 5v - 1 )\n- ( 4v + 7 )", "The average of these three values is computed by summing them and dividing by 3:", "[\n\ ext{Average} = \frac{(3v + 2) + (5v - 1) + (4v + 7)}{3}\n]", "Combine like terms in the numerator:", "[\n(3v + 5v + 4v) + (2 - 1 + 7) = 12v + 8\n]", "So the average becomes:", "[\n\ ext{Average} = \frac{12v + 8}{3}\n]", "---", "### Step 2: Apply the Condition — The Average Must Be Less Than 25", "We are told the average must be less than 25:", "[\n\frac{12v + 8}{3} < 25\n]", "Multiply both sides by 3 to eliminate the denominator:", "[\n12v + 8 < 75\n]", "Subtract 8 from both sides:", "[\n12v < 67\n]", "Divide both sides by 12:", "[\nv < \frac{67}{12}\n]", "Convert ( \frac{67}{12} ) to a decimal for clarity:", "[\n\frac{67}{12} = 5.583\overline{3}\n]", "Since ( v ) must be a positive integer, the largest integer less than 5.583… is:", "[\nv = 5\n]", "---", "### Step 3: Verify That ( v = 5 ) Satisfies All Conditions", "Let’s substitute ( v = 5 ) into each expression:", "- ( 3v + 2 = 3(5) + 2 = 15 + 2 = 17 )\n- ( 5v - 1 = 5(5) - 1 = 25 - 1 = 24 )\n- ( 4v + 7 = 4(5) + 7 = 20 + 7 = 27 )", "Sum: ( 17 + 24 + 27 = 68 )\nAverage: ( \frac{68}{3} \approx 22.67 ), which is indeed less than 25.", "Try ( v = 6 ) to confirm it violates the condition:", "- ( 3v + 2 = 20 )\n- ( 5v - 1 = 29 )\n- ( 4v + 7 = 31 )\nSum: 80 → Average: ( 80/3 \approx 26.67 > 25 ) → Too large.", "Thus, ( v = 5 ) is the maximum valid integer.", "---", "### Why This Approach Matters", "Understanding how to calculate averages of algebraic expressions secures your foundation in algebra. More importantly, applying inequalities—especially with integer constraints—turns abstract math into real-world problem-solving. Whether you're analyzing data trends, optimizing performance, or solving academic challenges, knowing how to set up and solve such inequalities is invaluable.", "---", "### Final Answer Recap", "The maximum positive integer value of ( v ) such that the average of ( 3v+2 ), ( 5v-1 ), and ( 4v+7 ) is less than 25 is ( \boxed{5} ).", "---", "Key Takeaways:", "- Always simplify expressions carefully before solving inequalities.\n- Apply arithmetic operations consistently (distribute, combine, divide appropriately).\n- Use exact fractional bounds and restrict to integers where required.\n- Verifying edge cases with test values ensures accuracy.", "Mastering these concepts empowers you not just to solve problems—like this one— but to think critically and apply math confidently in any context."]
