To determine how far Joey was from home after walking, we can break it down step by step.
- Joey walked east for \(15 \frac{2}{3}\) meters.
- He then walked west for \(21 \frac{3}{4}\) meters.
First, we need to convert both mixed numbers to improper fractions for easier calculation.
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Convert \(15 \frac{2}{3}\) to an improper fraction: \[ 15 \frac{2}{3} = \frac{15 \times 3 + 2}{3} = \frac{45 + 2}{3} = \frac{47}{3} \]
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Convert \(21 \frac{3}{4}\) to an improper fraction: \[ 21 \frac{3}{4} = \frac{21 \times 4 + 3}{4} = \frac{84 + 3}{4} = \frac{87}{4} \]
Next, to find out how far Joey is from home, we will subtract the distance he traveled east from the distance he traveled west (since east is positive and west is negative).
The total distance from home can be calculated as: \[ \text{Distance from home} = \left(\frac{87}{4} - \frac{47}{3}\right) \]
To perform this subtraction, we need a common denominator. The least common multiple of 4 and 3 is 12.
Convert each fraction:
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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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Convert \(\frac{47}{3}\) to a denominator of 12: \[ \frac{47}{3} = \frac{47 \times 4}{3 \times 4} = \frac{188}{12} \]
Now perform the subtraction: \[ \frac{261}{12} - \frac{188}{12} = \frac{261 - 188}{12} = \frac{73}{12} \]
Now convert \(\frac{73}{12}\) back to a mixed number.
Perform the division: 73 divided by 12 equals 6 with a remainder of 1.
Thus, \[ \frac{73}{12} = 6 \frac{1}{12} \]
So, Joey was \(6 \frac{1}{12}\) meters from home.
Final answer: Joey was 6 1/12 meters from home.
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