To solve for \( d \), find a common denominator for the fractions, which is 240:

["How to Solve for ( d ) by Finding a Common Denominator: A Step-by-Step Guide (Including a Denominator of 240)", "Understanding how to solve equations involving fractions is a fundamental skill in algebra. One common technique is finding a common denominator—often a multiple like 240—for efficient addition, subtraction, or simplification. In many word problems or mathematical exercises, expressing fractions with a common denominator simplifies solving for variables like ( d ). This article breaks down how to solve for ( d ) by finding a common denominator of 240 and demonstrates the process clearly.", "### Why Find a Common Denominator?", "Fractions with different denominators cannot be added or subtracted directly. By converting fractions to equivalent forms with a shared denominator, you align their parts, allowing accurate arithmetic operations. Choosing 240 as the common denominator is especially useful because it’s a multiple of many common denominators (like 12, 15, 16, etc.), making calculations with multiples and simplifications easier.", "### Scenario: Solving for ( d ) in a Fraction Equation", "Let’s consider a typical problem where solving for ( d ) requires combining or isolating this variable via fractions. Suppose we’re given the equation:", "[\n\frac{7}{24}d - \frac{5}{16} = \frac{1}{240}\n]", "Our goal is to isolate ( d ), and a common denominator of 240 plays a key role in this process.", "### Step 1: Understand the Denominators", "Identify all denominators:\n- Denominator 1: 24\n- Denominator 2: 16\n- Target denominator: 240", "Confirm 240 is a common multiple:\n( \ ext{LCM}(24, 16, 240) = 240 ), so 240 works perfectly.", "### Step 2: Convert Each Fraction to Have Denominator 240", "Convert each fraction by multiplying numerator and denominator by a factor to reach 240:", "- For ( \frac{7}{24} ):\n ( 240 \div 24 = 10 ) → Multiply numerator and denominator by 10\n [\n \frac{7}{24} = \frac{7 \ imes 10}{24 \ imes 10} = \frac{70}{240}\n ]", "- For ( \frac{5}{16} ):\n ( 240 \div 16 = 15 ) → Multiply numerator and denominator by 15\n [\n \frac{5}{16} = \frac{5 \ imes 15}{16 \ imes 15} = \frac{75}{240}\n ]", "- The right-hand side:\n ( \frac{1}{240} ) already has the desired denominator.", "Now rewrite the original equation:", "[\n\frac{70}{240}d - \frac{75}{240} = \frac{1}{240}\n]", "### Step 3: Eliminate the Denominator", "Since all terms have denominator 240, multiply every term by 240 to eliminate fractions:", "[\n240 \cdot \left( \frac{70}{240}d \right) - 240 \cdot \left( \frac{75}{240} \right) = 240 \cdot \left( \frac{1}{240} \right)\n]", "This simplifies cleanly:", "[\n70d - 75 = 1\n]", "### Step 4: Solve the Simplified Equation", "Add 75 to both sides:", "[\n70d = 1 + 75 = 76\n]", "Now divide both sides by 70:", "[\nd = \frac{76}{70}\n]", "Reduce the fraction:", "[\nd = \frac{38}{35}\n]", "### Step 5: Final Answer", "[\n\boxed{d = \frac{38}{35}}\n]", "### Why This Method Works", "By standardizing all fractions to a common denominator—here 240—we eliminate complexity, avoid messy multiplications, and make algebraic manipulation straightforward. This technique is especially valuable in timed tests, homework, and real-world applications where precision and clarity matter.", "### Summary", "Finding a common denominator—like 240—streamlines solving for unknowns in equations with fractions. This method enhances accuracy, simplifies arithmetic, and prepares you for more complex algebraic challenges. Whether calculating rates, proportions, or physics problems, mastering this step is essential.", "---", "Keywords: solve for ( d ), common denominator, fractions, algebra, find 240, LCM, dimensional analysis, mathematical equation solving, reduce fraction, math tutorial."]