To find out how far Joey is from home after walking east and then west, we begin by calculating the total distance he walked in each direction.
- Joey walked east: \( 15 \frac{2}{3} \) meters.
- Joey walked west: \( 21 \frac{3}{4} \) meters.
First, we convert the mixed numbers into improper fractions to make calculations easier.
For \( 15 \frac{2}{3} \): \[ 15 \frac{2}{3} = \frac{15 \times 3 + 2}{3} = \frac{45 + 2}{3} = \frac{47}{3} \]
For \( 21 \frac{3}{4} \): \[ 21 \frac{3}{4} = \frac{21 \times 4 + 3}{4} = \frac{84 + 3}{4} = \frac{87}{4} \]
Next, we need to find the total distance Joey is from home after both walks. Since he walked east first and then west, we subtract the eastward distance from the westward distance:
\[ \text{Total distance from home} = \text{Distance walked west} - \text{Distance walked east} \] \[ \text{Total distance from home} = \frac{87}{4} - \frac{47}{3} \]
To perform this subtraction, we first need a common denominator. The least common multiple of 4 and 3 is 12.
Now we convert each fraction: \[ \frac{87}{4} = \frac{87 \times 3}{4 \times 3} = \frac{261}{12} \] \[ \frac{47}{3} = \frac{47 \times 4}{3 \times 4} = \frac{188}{12} \]
Now we subtract the two fractions: \[ \frac{261}{12} - \frac{188}{12} = \frac{261 - 188}{12} = \frac{73}{12} \]
Finally, we convert \( \frac{73}{12} \) back to a mixed number: \[ 73 \div 12 = 6 \quad \text{(remainder 1)} \] Thus, \( \frac{73}{12} = 6 \frac{1}{12} \).
Joey was \( 6 \frac{1}{12} \) meters from home.
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