To determine Joey's distance from home, we need to find the difference between the distances he walked away from home and the distance he walked back towards home.
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First, let’s denote Joey's distances:
- Distance walked away from home: \( \frac{21}{34} \) meters
- Distance walked back towards home: \( \frac{15}{23} \) meters
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To find out how far he is from home, we need to compute: \[ \text{Distance from home} = \frac{21}{34} - \frac{15}{23} \]
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To subtract these fractions, we first find a common denominator. The denominators are 34 and 23. The least common multiple (LCM) of 34 and 23 is \( 34 \times 23 = 782 \).
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Now we convert each fraction to have this common denominator:
- For \( \frac{21}{34} \): \[ \frac{21}{34} = \frac{21 \times 23}{34 \times 23} = \frac{483}{782} \]
- For \( \frac{15}{23} \): \[ \frac{15}{23} = \frac{15 \times 34}{23 \times 34} = \frac{510}{782} \]
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Now we can perform the subtraction: \[ \frac{21}{34} - \frac{15}{23} = \frac{483}{782} - \frac{510}{782} = \frac{483 - 510}{782} = \frac{-27}{782} \]
Since the result is negative, this indicates that Joey walked back more than he walked away from home:
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Compute the absolute value to find the distance from home: \[ \left| \frac{-27}{782} \right| = \frac{27}{782} \]
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Now we convert \( \frac{27}{782} \) into a mixed number.
- 27 is less than 782, so it is already a proper fraction. The mixed number representation is simply \( 0 \frac{27}{782} \).
Hence, as a mixed number, Joey is \( \frac{27}{782} \) meters from home.
Final answer: \[ \frac{27}{782} \]