To determine the constant of variation from the given table, follow these detailed steps:
1. Understand the relationship given in the table: We are given the time in minutes (denoted as [tex]\( x \)[/tex]) and the corresponding number of pages printed (denoted as [tex]\( y \)[/tex]).
2. Identify the pairs from the table:
- For [tex]\( x = 2 \)[/tex] minutes, [tex]\( y = 3 \)[/tex] pages.
- For [tex]\( x = 6 \)[/tex] minutes, [tex]\( y = 9 \)[/tex] pages.
- For [tex]\( x = 8 \)[/tex] minutes, [tex]\( y = 12 \)[/tex] pages.
- For [tex]\( x = 18 \)[/tex] minutes, [tex]\( y = 27 \)[/tex] pages.
3. Calculate the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair to understand the consistency of the ratio, which will help in finding the constant of variation:
- When [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{3}{2} = 1.5
\][/tex]
- When [tex]\( x = 6 \)[/tex] and [tex]\( y = 9 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{9}{6} = 1.5
\][/tex]
- When [tex]\( x = 8 \)[/tex] and [tex]\( y = 12 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{12}{8} = 1.5
\][/tex]
- When [tex]\( x = 18 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{27}{18} = 1.5
\][/tex]
4. Verify if the ratio is constant across all pairs:
Since in all cases the ratio [tex]\( \frac{y}{x} = 1.5 \)[/tex], we can conclude that the ratio is indeed consistent.
5. Determine the constant of variation:
The constant of variation (k) is the consistent ratio we found.
Hence, the constant of variation is [tex]\( 1.5 \)[/tex].
Given the available choices:
[tex]\[
\frac{2}{3}, \quad \frac{3}{2}, \quad 2, \quad 3
\][/tex]
### Answer:
The constant of variation matches [tex]\( \frac{3}{2} \)[/tex], which is equivalent to 1.5. Therefore, the correct answer is [tex]\(\boxed{\frac{3}{2}}\)[/tex].
