The sum of the reciprocals of two numbers is \( \frac{11}{30} \). If one number is 5, find the other.
["Title: Solve for the Missing Number: Sum of Reciprocals Equation with One Value Known", "When faced with the equation involving reciprocals of two numbers, many learners feel overwhelmed—but with a clear method, solving it becomes simple. In this article, we tackle a classic problem: The sum of the reciprocals of two numbers is ( \frac{11}{30} ). If one number is 5, find the other.", "---", "### Understanding the Problem", "We are given:", "[\n\frac{1}{x} + \frac{1}{5} = \frac{11}{30}\n]", "Our goal is to find the value of ( x ). This problem exemplifies a real-world application in algebra where reciprocals play a key role—common in rates, proportions, and engineering calculations.", "---", "### Step 1: Isolate the Unknown Reciprocal", "Start by subtracting ( \frac{1}{5} ) from both sides:", "[\n\frac{1}{x} = \frac{11}{30} - \frac{1}{5}\n]", "We need a common denominator to subtract the fractions. The least common denominator of 30 and 5 is 30.", "Convert ( \frac{1}{5} ) to thirtieth:", "[\n\frac{1}{5} = \frac{6}{30}\n]", "Now substitute:", "[\n\frac{1}{x} = \frac{11}{30} - \frac{6}{30} = \frac{5}{30} = \frac{1}{6}\n]", "---", "### Step 2: Solve for ( x )", "We now have:", "[\n\frac{1}{x} = \frac{1}{6}\n]", "Taking reciprocals on both sides (or recognizing inequality of reciprocals: smaller denominator means larger fraction):", "[\nx = 6\n]", "---", "### Final Answer", "The missing number is ( 6 ). Verifying:", "[\n\frac{1}{6} + \frac{1}{5} = \frac{5 + 6}{30} = \frac{11}{30}\n]", "✔️ Correct and confirmed.", "---", "### Why This Problem Matters", "Understanding how to work with sums of reciprocals builds a foundation for solving rational equations, which appear frequently in science, finance, and optimization problems. Being able to isolate and manipulate reciprocals is a valuable skill.", "---", "Bottom Line:\nIf the sum of reciprocals of two numbers is ( \frac{11}{30} ), and one number is 5, the other is clearly 6.", "---", "Keywords for SEO:\nsum of reciprocals equation, solve reciprocal sum, find missing number in reciprocal sum, algebra problem solution, reciprocal sum with known number 5, how to find second number when sum of reciprocals is 11/30", "Meta Description:\nGiven the sum of reciprocals of two numbers is ( \frac{11}{30} ) and one number is 5, find the missing number. Step-by-step solution with verification included."]
