To find the constant of proportionality from \( y \) to \( z \), we can use the formula for proportional relationships given by:
\[ k = \frac{z}{y} \]
Where \( k \) is the constant of proportionality.
We can use any of the given pairs \( (y, z) \) to find \( k \). Let's calculate \( k \) using each pair:
- Using the first pair \( (72, 88) \):
\[ k = \frac{88}{72} = \frac{88 \div 8}{72 \div 8} = \frac{11}{9} \]
- Using the second pair \( (99, 111) \):
\[ k = \frac{111}{99} = \frac{111 \div 3}{99 \div 3} = \frac{37}{33} \]
- Using the third pair \( (153153, 1717) \):
\[ k = \frac{1717}{153153} \]
To simplify \( \frac{1717}{153153} \), we can first find their greatest common divisor. We can break them down:
- Factor \( 1717 = 17 \times 101 \).
- Factor \( 153153 \). To find the factors, we can test small primes.
After factorization, we found: \[ 153153 = 3 \times 7 \times 11 \times 13 \times 17 \] Thus, we can simplify: \[ \frac{1717}{153153} = \frac{17 \times 101}{3 \times 7 \times 11 \times 13 \times 17} = \frac{101}{3 \times 7 \times 11 \times 13} = \frac{101}{3003} \]
Thus, our pairs yield different proportional constants, indicating that it might not be a single constant throughout.
To summarize:
- From \( (72, 88) \), \( k = \frac{11}{9} \)
- From \( (99, 111) \), \( k = \frac{37}{33} \)
- From \( (153153, 1717) \), \( k = \frac{101}{3003} \)
Conclusively, the constant of proportionality is not uniform across all pairs. Checking these for consistency, can be crucial if these were to be parts of a proportional relationship system, we normally expect a constant result; all these suggest variability. Hence there is no single constant of proportionality across all pairs. If you need one, you should state the conditions for the relationship, based on these values.