To find out how far Joey is from home after his walk, we need to calculate his total displacement by considering the distances he walked in each direction.
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Walking East: Joey walked \(15 \frac{2}{3}\) meters. To convert this to an improper fraction: \[ 15 \frac{2}{3} = 15 + \frac{2}{3} = \frac{15 \times 3 + 2}{3} = \frac{45 + 2}{3} = \frac{47}{3} \text{ meters} \]
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Walking West: Then, Joey walked \(21 \frac{3}{4}\) meters back. Converting this to an improper fraction: \[ 21 \frac{3}{4} = 21 + \frac{3}{4} = \frac{21 \times 4 + 3}{4} = \frac{84 + 3}{4} = \frac{87}{4} \text{ meters} \]
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Calculating the Total Displacement:
- When Joey walks East (positive direction), we add the distance.
- When he walks West (negative direction), we subtract the distance.
So, the total displacement from home can be calculated as: \[ \text{Total Displacement} = \text{East} - \text{West} = \frac{47}{3} - \frac{87}{4} \]
To perform this subtraction, we need a common denominator. The least common multiple of \(3\) and \(4\) is \(12\).
Converting both fractions to have a denominator of \(12\):
- For \(\frac{47}{3}\): \[ \frac{47}{3} = \frac{47 \times 4}{3 \times 4} = \frac{188}{12} \]
- For \(\frac{87}{4}\): \[ \frac{87}{4} = \frac{87 \times 3}{4 \times 3} = \frac{261}{12} \]
Now we can subtract: \[ \text{Total Displacement} = \frac{188}{12} - \frac{261}{12} = \frac{188 - 261}{12} = \frac{-73}{12} \]
The negative sign indicates that Joey is \(73/12\) meters to the WEST of home.
- Converting the Displacement to Mixed Number: \[ 73 \div 12 = 6 \text{ R } 1 \] Thus, \(\frac{73}{12} = 6 \frac{1}{12}\) meters.
So, after all calculations, Joey is 6 \( \frac{1}{12} \) meters from home, and specifically, he is 6 \( \frac{1}{12} \) meters WEST of his starting point.
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