To find out how far Joey is from home, we need to calculate his total movement to the east and west.
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Convert the mixed numbers to improper fractions:
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For \( 15 \frac{2}{3} \): \[ 15 \frac{2}{3} = \frac{15 \times 3 + 2}{3} = \frac{45 + 2}{3} = \frac{47}{3} \]
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For \( 21 \frac{3}{4} \): \[ 21 \frac{3}{4} = \frac{21 \times 4 + 3}{4} = \frac{84 + 3}{4} = \frac{87}{4} \]
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Convert these fractions to a common denominator to make it easier to compute:
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The least common multiple of 3 and 4 is 12.
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Convert \( \frac{47}{3} \) to a denominator of 12: \[ \frac{47}{3} \times \frac{4}{4} = \frac{188}{12} \]
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Convert \( \frac{87}{4} \) to a denominator of 12: \[ \frac{87}{4} \times \frac{3}{3} = \frac{261}{12} \]
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Now, subtract the distance walked west from the distance walked east: \[ \text{Total distance from home} = \frac{188}{12} - \frac{261}{12} = \frac{188 - 261}{12} = \frac{-73}{12} \]
The negative sign indicates that Joey is west of his starting point.
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Convert \( \frac{73}{12} \) to a mixed number:
- \(\frac{73}{12}\) can be calculated as:
- \(73\) divided by \(12\) is \(6\) with a remainder of \(1\).
- Therefore, \( \frac{73}{12} = 6 \frac{1}{12} \).
- \(\frac{73}{12}\) can be calculated as:
Joey is \( 6 \frac{1}{12} \) meters away from home, in the west direction. Thus, the answer is:
\[ \text{Joey is } 6 \frac{1}{12} \text{ meters from home.} \]
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