Joey walked away from home for 21 3/4 meters. Then, he walked back in the direction of his home for 15 2/3

To find out how far Joey is from home after walking away and then back, we need to perform the following calculations:

1. Convert the mixed numbers into improper fractions to make calculations easier.

For \( 21 \frac{3}{4} \):
\[
21 \frac{3}{4} = \frac{21 \times 4 + 3}{4} = \frac{84 + 3}{4} = \frac{87}{4}
\]

For \( 15 \frac{2}{3} \):
\[
15 \frac{2}{3} = \frac{15 \times 3 + 2}{3} = \frac{45 + 2}{3} = \frac{47}{3}
\]

2. Joey walked away from home for \( \frac{87}{4} \) meters and then returned \( \frac{47}{3} \) meters.

3. To find out how far Joey is from home, we need to subtract the distance he walked back from the distance he walked away:

\[
\text{Distance from home} = \frac{87}{4} - \frac{47}{3}
\]

4. To subtract these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12.

Convert \( \frac{87}{4} \) to a fraction with a denominator of 12:
\[
\frac{87}{4} \times \frac{3}{3} = \frac{261}{12}
\]

Convert \( \frac{47}{3} \) to a fraction with a denominator of 12:
\[
\frac{47}{3} \times \frac{4}{4} = \frac{188}{12}
\]

5. Now perform the subtraction:
\[
\frac{261}{12} - \frac{188}{12} = \frac{261 - 188}{12} = \frac{73}{12}
\]

6. Convert \( \frac{73}{12} \) back into a mixed number:
\[
73 \div 12 = 6 \quad \text{(remainder \( 1 \))}
\]
Thus, \( \frac{73}{12} = 6 \frac{1}{12} \).

Therefore, Joey is \( 6 \frac{1}{12} \) meters away from home.