Question

The table below shows a proportional relationship between ss and tt.
ss tt
5454 99
108108 1818
120120 2020
Find the constant of proportionality from ss to tt. Express your answer as a fraction in reduced terms.

Answer 1

To find the constant of proportionality from \( ss \) to \( tt \), we need to divide each \( tt \) value by its corresponding \( ss \) value.

We can do this for any of the given pairs. Let's calculate the constant of proportionality using the first pair (5454, 99):

\[
k = \frac{tt}{ss} = \frac{99}{5454}
\]

Next, we simplify this fraction.

1. First, we need to find the greatest common divisor (GCD) of 99 and 5454. We can start by factoring both numbers:

- The prime factorization of 99 is:
\[
99 = 3^2 \times 11
\]

- The prime factorization of 5454 is:
\[
5454 = 2 \times 3^3 \times 11
\]

2. To find the GCD, we identify the minimum power of each common factor:
- For \( 3 \): minimum power is \( 2 \).
- For \( 11 \): minimum power is \( 1 \).

Therefore, the GCD is:
\[
GCD = 3^2 \times 11 = 9 \times 11 = 99
\]

3. Now we can divide both the numerator and denominator of \( \frac{99}{5454} \) by 99:

\[
\frac{99 \div 99}{5454 \div 99} = \frac{1}{55.0} = \frac{1}{55}
\]

Thus, the constant of proportionality \( k \) from \( ss \) to \( tt \) is:

\[
\boxed{\frac{1}{55}}
\]