Let's analyze Leonardo's work step by step to identify where the error occurred.
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Starting Equation: \[ 4 \left( x - \frac{1}{5} \right) = 2 \frac{2}{3} \]
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Convert Mixed Number to Improper Fraction: \[ 2 \frac{2}{3} = \frac{8}{3} \] So the equation is: \[ 4 \left( x - \frac{1}{5} \right) = \frac{8}{3} \]
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Step 1: Distributing the 4: \[ 4x - 4 \cdot \frac{1}{5} = \frac{8}{3} \] This simplifies to: \[ 4x - \frac{4}{5} = \frac{8}{3} \]
- This step is correct.
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Step 2: Adding \(\frac{4}{5}\) to both sides: \[ 4x = \frac{8}{3} + \frac{4}{5} \]
To add \(\frac{8}{3}\) and \(\frac{4}{5}\), we need a common denominator. The least common multiple of 3 and 5 is 15.
Convert each fraction: \[ \frac{8}{3} = \frac{8 \cdot 5}{3 \cdot 5} = \frac{40}{15} \] \[ \frac{4}{5} = \frac{4 \cdot 3}{5 \cdot 3} = \frac{12}{15} \] So, \[ 4x = \frac{40}{15} + \frac{12}{15} = \frac{52}{15} \]
- This is where Leonardo makes an error. He wrote \(4x = \frac{56}{15}\) instead of \(\frac{52}{15}\).
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Step 3: Dividing by 4 to solve for \(x\): Correct calculation based on the correct equation: \[ x = \frac{52}{15} \cdot \frac{1}{4} = \frac{52}{60} = \frac{13}{15} \] Leonardo incorrectly calculated \(x = \frac{14}{15}\).
The error is in Step 2, where he incorrectly added \(\frac{8}{3} + \frac{4}{5}\) and ended up with \(\frac{56}{15}\) instead of the correct sum, which is \(\frac{52}{15}\).
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