To find out how far Joey is from home, we need to calculate the total distance he walked in each direction and determine his final position relative to home.
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First, let's convert the mixed numbers into 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, we convert the distances he walked east and west to have a common denominator so we can subtract them easily. The least common multiple of 3 and 4 is 12.
Convert \( \frac{47}{3} \) to a denominator of 12: \[ \frac{47}{3} = \frac{47 \times 4}{3 \times 4} = \frac{188}{12} \]
Convert \( \frac{87}{4} \) to a denominator of 12: \[ \frac{87}{4} = \frac{87 \times 3}{4 \times 3} = \frac{261}{12} \]
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Now, subtract the distance walked west from the distance walked east: \[ \frac{188}{12} - \frac{261}{12} = \frac{188 - 261}{12} = \frac{-73}{12} \]
The negative sign indicates that Joey walked more west than east. Thus, we take the absolute value: \[ \left| \frac{-73}{12} \right| = \frac{73}{12} \]
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Now, convert \( \frac{73}{12} \) back to a mixed number:
- Divide 73 by 12.
- The whole number part is \( 6 \) (since \( 12 \times 6 = 72 \)).
- The remainder is \( 1 \) (since \( 73 - 72 = 1 \)).
- Thus, \( \frac{73}{12} = 6 \frac{1}{12} \).
So, Joey was \( 6 \frac{1}{12} \) meters from home.
Final answer: Joey was \( 6 \frac{1}{12} \) meters from home.
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