To find out how far Joey is from home after walking east and then west, we first need to convert the mixed numbers into improper fractions for easier calculation.
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Convert 15 2/3 to an improper fraction: \[ 15 2/3 = 15 + \frac{2}{3} = \frac{15 \times 3 + 2}{3} = \frac{45 + 2}{3} = \frac{47}{3} \]
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Convert 21 3/4 to an improper fraction: \[ 21 3/4 = 21 + \frac{3}{4} = \frac{21 \times 4 + 3}{4} = \frac{84 + 3}{4} = \frac{87}{4} \]
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Joey walked east, which is positive, and then west, which is negative:
- The total distance Joey is from home can be calculated as: \[ \text{Distance from home} = \left(\text{Distance East}\right) - \left(\text{Distance West}\right) \] This translates to: \[ \text{Distance from home} = \frac{47}{3} - \frac{87}{4} \]
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To subtract these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12.
Convert \(\frac{47}{3}\) to a fraction with a denominator of 12: \[ \frac{47}{3} = \frac{47 \times 4}{3 \times 4} = \frac{188}{12} \]
Convert \(\frac{87}{4}\) to a fraction with a denominator of 12: \[ \frac{87}{4} = \frac{87 \times 3}{4 \times 3} = \frac{261}{12} \]
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Now subtract the two fractions: \[ \frac{188}{12} - \frac{261}{12} = \frac{188 - 261}{12} = \frac{-73}{12} \]
Since distance cannot be negative, we take the absolute value: \[ \text{Distance from home} = \frac{73}{12} \]
- Convert \(\frac{73}{12}\) to a mixed number:
- Divide 73 by 12: \[ 73 \div 12 = 6 \quad \text{(whole number part)} \]
- The remainder is: \[ 73 - (12 \times 6) = 73 - 72 = 1 \]
Thus, we can write: \[ \frac{73}{12} = 6 \frac{1}{12} \]
So, Joey is \( \boxed{6 \frac{1}{12}} \) meters from home.