To find out how far Joey is from home, we can start by converting the mixed numbers into improper fractions and then perform the calculations.
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Convert the mixed numbers to improper fractions.
- For \(15 \frac{2}{3}\): \[ 15 \frac{2}{3} = \frac{15 \times 3 + 2}{3} = \frac{45 + 2}{3} = \frac{47}{3} \]
- 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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Now, Joey walks east 15 2/3 meters (which is \(\frac{47}{3}\)) and then west 21 3/4 meters (which is \(\frac{87}{4}\)).
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To find the total distance from home, we will subtract the distance walked west from the distance walked east: \[ \text{Distance from home} = \text{Distance east} - \text{Distance west} \] We need a common denominator. The least common multiple of 3 and 4 is 12.
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Convert 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 \(\frac{87}{4}\): \[ \frac{87}{4} = \frac{87 \times 3}{4 \times 3} = \frac{261}{12} \]
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Now we can subtract: \[ \text{Distance from home} = \frac{188}{12} - \frac{261}{12} = \frac{188 - 261}{12} = \frac{-73}{12} \]
Since the distance should be positive, we take the absolute value: \[ \text{Distance from home} = \frac{73}{12} \]
- Finally, convert \(\frac{73}{12}\) back into a mixed number: \[ 73 \div 12 = 6 \quad \text{R} 1 \quad \Rightarrow \quad 6 \frac{1}{12} \]
So, Joey was \(6 \frac{1}{12}\) meters from home.