To determine how far Joey is from home, we need to calculate the total distance he walked in each direction, then find the net distance from home.
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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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Joey walked east \(15 \frac{2}{3}\) m and then walked west \(21 \frac{3}{4}\) m. To find the net distance from home, we will subtract the distance traveled east from the distance traveled west: \[ \text{Net distance} = \text{Distance walked west} - \text{Distance walked east} \] \[ \text{Net distance} = \frac{87}{4} - \frac{47}{3} \]
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To subtract these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12.
Convert both fractions:
- For \(\frac{87}{4}\): \[ \frac{87}{4} = \frac{87 \times 3}{4 \times 3} = \frac{261}{12} \]
- For \(\frac{47}{3}\): \[ \frac{47}{3} = \frac{47 \times 4}{3 \times 4} = \frac{188}{12} \]
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Now we can perform the subtraction: \[ \text{Net distance} = \frac{261}{12} - \frac{188}{12} = \frac{261 - 188}{12} = \frac{73}{12} \]
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Convert the improper fraction to a mixed number:
- Divide 73 by 12 to find the whole number part: \[ 73 \div 12 = 6 \quad \text{(remainder 1)} \] So, \(73 = 6 \times 12 + 1\).
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Therefore, the mixed number is: \[ 6 \frac{1}{12} \]
Joey is \(6 \frac{1}{12}\) m from home.