["# Solving the Equation × 1.22 = 280 × (122/100) Step-by-Step", "Understanding how to solve equations involving decimals and fractions is essential in math and real-world applications, from finance to science. This guide breaks down the equation × ×1.22 = 280 × (122/100) and shows that the solution equals 341.6, confirming the correctness of the calculation.", "---", "## Understanding the Equation", "We start with the equation:", "[
\nx \ imes 1.22 = 280 \ imes \left( \frac{122}{100} \right)
\n]", "Here, the unknown variable ( x ) must be isolated to determine its exact value. This involves basic algebraic manipulation and arithmetic with decimals and fractions.", "---", "## Step 1: Simplify the Right Side", "First, convert the fraction ( \frac{122}{100} ) into a decimal:", "[
\n\frac{122}{100} = 1.22
\n]", "Now, substitute back into the equation:", "[
\nx \ imes 1.22 = 280 \ imes 1.22
\n]", "---", "## Step 2: Multiply 280 × 1.22", "Calculate the right-hand side:", "[
\n280 \ imes 1.22 = 341.60
\n]", "(Note: ( 280 \ imes 1.22 = 280 \ imes \frac{122}{100} = \frac{280 \ imes 122}{100} = \frac{34160}{100} = 341.6 ))", "---", "## Step 3: Solve for ( x )", "Now the equation is:", "[
\nx \ imes 1.22 = 341.6
\n]", "To isolate ( x ), divide both sides by 1.22:", "[
\nx = \frac{341.6}{1.22}
\n]", "But since ( 340 \div 1.22 ) simplifies neatly:", "[
\nx = \frac{341.6}{1.22} = 341.6 \div 1.22
\n]", "Multiply numerator and denominator by 100 to eliminate decimals:", "[
\nx = \frac{34160}{122}
\n]", "---", "## Step 4: Final Simplification", "Perform the division:", "[
\n34160 \div 122 = 341.60 \div 1.22 = 281? \quad \ ext{Wait, actually:", "Actually, let’s check:", "[
\n\frac{34160}{122} = 281 \quad \ ext{(since } 122 \ imes 281 = 34162 \ ext{ — close!)}
\n]", "Wait — correction: precise calculation:", "Try direct division:", "[
\n341.60 \div 1.22 = ?
\n]", "Multiply numerator and denominator by 100:", "[
\n\frac{34160}{122} = ?
\n]", "Now simplify:", "[
\n34160 \div 122
\n]", "Break down:", "[
\n122 \ imes 280 = 34160 \quad \ ext{(Yes!)}
\n]", "So:", "[
\n\frac{34160}{122} = 280
\n]", "Wait — this contradicts earlier result? No — let’s reconcile.", "Wait: the original equation was:", "[
\nx \ imes 1.22 = 280 \ imes \frac{122}{100} = 280 \ imes 1.22 = 341.6
\n]", "Thus:", "[
\nx = \frac{341.6}{1.22} = ?
\n]", "But ( 341.6 \div 1.22 = 281 )? Let's compute:", "[
\n1.22 \ imes 281 = 1.22 \ imes 200 = 244, \quad 1.22 \ imes 80 = 97.6, \quad 1.22 \ imes 1 = 1.22
\n\Rightarrow 244 + 97.6 + 1.22 = 342.82 \quad (\ ext{too high})
\n]", "Better: use calculator-style math:", "[
\n341.6 \div 1.22 = ?
\n]", "Multiply numerator and denominator by 100:", "[
\n\frac{34160}{122} = ?
\n]", "Now simplify fraction:", "Find GCD of 34160 and 122.
\n122 = 2 × 61
\n34160 ÷ 122:", "[
\n122 \ imes 280 = 34160 \quad \ ext{exactly!}
\n]", "So:", "[
\n\frac{34160}{122} = 280
\n]", "Wait — contradiction? No — wait: original problem says:", "[
\nx \ imes 1.22 = 280 \ imes \frac{122}{100} = \frac{280 \ imes 122}{100} = \frac{34160}{100} = 341.6
\n]", "So:", "[
\nx \ imes 1.22 = 341.6
\n\Rightarrow x = \frac{341.6}{1.22}
\n]", "Now compute:", "[
\n\frac{341.6}{1.22} = \frac{34160}{122} = 280 \quad \ ext{?}
\n]", "Wait — this only equals 280 if:", "[
\n\frac{341.6}{1.22} = 280 \quad \ ext{because } 1.22 \ imes 280 = 341.6
\n]", "Yes, that’s correct:", "[
\n1.22 \ imes 280 = (122/100) \ imes 280 = \frac{122 \ imes 280}{100} = \frac{34160}{100} = 341.6
\n]", "Hence:", "[
\nx = \frac{341.6}{1.22} = 280
\n]", "Wait — but earlier step said ( \frac{280 \ imes 122}{100} = 341.6 ), so:", "[
\nx = \frac{280 \ imes 122 / 100}{1.22} = \frac{341.6}{1.22} = 280
\n]", "But that would mean:", "[
\nx \ imes 1.22 = 280 \ imes 1.22 = 341.6 \Rightarrow x = 280
\n]", "Yes — so ( x = 280 ) is the solution.", "But wait — original equation was:", "[
\nx \ imes 1.22 = 280 \ imes 1.22 = 341.6
\n\Rightarrow x = 341.6 \div 1.22 = 280
\n]", "So final answer is:", "[
\nx = 280
\n]", "But earlier step said:", "[
\nx = \frac{341.6}{1.22} = 280
\n]", "Yes — so corrected:", "---", "## ✅ Final Calculation Summary", "1. Evaluate right-hand side:
\n [
\n 280 \ imes \frac{122}{100} = 280 \ imes 1.22 = 341.6
\n ]", "2. Set up equation:
\n [
\n x \ imes 1.22 = 341.6
\n ]", "3. Solve for ( x ):
\n [
\n x = \frac{341.6}{1.22} = 280
\n ]", "Alternatively, recognizing the simplification:", "[
\nx = \frac{280 \ imes \frac{122}{100}}{1.22} = \frac{280 \ imes 1.22}{1.22} = 280
\n]", "---", "## Why This Method Works", "Using fractions avoids decimal errors:", "[
\nx \ imes 1.22 = 280 \ imes \frac{122}{100} \Rightarrow x = \frac{280 \ imes \frac{122}{100}}{1.22} = 280 \ imes \frac{122}{100 \ imes 1.22}
\n]", "Since ( 100 \ imes 1.22 = 122 ), we get:", "[
\nx = 280 \ imes \frac{122}{122} = 280 \ imes 1 = 280
\n]", "---", "## Real-World Example", "Imagine calculating a price increase:
\nA product increased by 22% to $341.60. What was the original price before the 1.22× multiplier?
\nIf the 1.22× factor equals ( 280 \ imes (122/100) ), then the original price is 280, and the multiplier explains the rest.", "---", "## Key Takeaways", "- Convert fractions and decimals early for clarity.
\n- Multiplication is associative — rearrange terms algebraically.
\n- Use fractions to avoid rounding errors from decimals.
\n- Always simplify fractions before performing division.", "---", "Final Answer:**
\n[
\n\boxed{340 \ imes 1.22 = 280 \ imes \frac{122}{100} = 341.6 \Rightarrow x = 280}
\n]", "The calculation is verified: × 1.22 = 280 × (122/100) confirms ( x = 280 ), and ( 280 \ imes 1.22 = 341.6 ), fully consistent."]
