To determine the constant of proportionality [tex]\( r \)[/tex] in the equation [tex]\( y = rx \)[/tex], we start by looking at the given pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values. We will use these pairs to find [tex]\( r \)[/tex].
Let's denote the pairs as follows:
- Pair 1: [tex]\( (x_1, y_1) = \left( 11, 1 + \frac{2}{9} \right) \)[/tex]
- Pair 2: [tex]\( (x_2, y_2) = \left( 21, 2 + \frac{1}{3} \right) \)[/tex]
- Pair 3: [tex]\( (x_3, y_3) = \left( 45, 5 \right) \)[/tex]
First, we convert the mixed numbers into improper fractions or decimal form to make the calculations easier:
- [tex]\( y_1 = 1 + \frac{2}{9} = 1 + 0.2222\overline{2} = 1.2222\overline{2} \)[/tex]
- [tex]\( y_2 = 2 + \frac{1}{3} = 2 + 0.3333\overline{3} = 2.3333\overline{3} \)[/tex]
- [tex]\( y_3 = 5 \)[/tex]
Now we calculate [tex]\( r \)[/tex] using each pair by dividing [tex]\( y \)[/tex] by [tex]\( x \)[/tex]:
1. For the first pair [tex]\((x_1, y_1)\)[/tex]:
[tex]\[
r_1 = \frac{y_1}{x_1} = \frac{1.2222\overline{2}}{11} = 0.11111111111111112
\][/tex]
2. For the second pair [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
r_2 = \frac{y_2}{x_2} = \frac{2.3333\overline{3}}{21} = 0.11111111111111112
\][/tex]
3. For the third pair [tex]\((x_3, y_3)\)[/tex]:
[tex]\[
r_3 = \frac{y_3}{x_3} = \frac{5}{45} = 0.1111111111111111
\][/tex]
Note that in a perfectly proportional relationship, [tex]\( r \)[/tex] should be constant across all pairs. Here, we can observe that:
- [tex]\( r_1 = 0.11111111111111112 \)[/tex]
- [tex]\( r_2 = 0.11111111111111112 \)[/tex]
- [tex]\( r_3 = 0.1111111111111111 \)[/tex]
The values [tex]\( r_1 \)[/tex], [tex]\( r_2 \)[/tex], and [tex]\( r_3 \)[/tex] are extremely close to each other, considering minor computational differences.
Therefore, the constant of proportionality [tex]\( r \)[/tex] is approximately:
[tex]\[
r = 0.1111111111111111
\][/tex]
So, [tex]\( r \approx 0.1111 \)[/tex] (rounded to four decimal places).
