To identify the variation in the given pair of values for [tex]\( X \)[/tex] and [tex]\( Y \)[/tex], we will follow these steps:
1. Calculate the ratio of [tex]\( X \)[/tex] to [tex]\( Y \)[/tex]: For each pair of values, [tex]\((X_i, Y_i)\)[/tex], we determine [tex]\( \frac{X_i}{Y_i} \)[/tex]. This helps us see if the relationship between [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] is consistent and whether they vary directly or inversely.
2. Compare the ratios: If the ratio [tex]\( \frac{X}{Y} \)[/tex] is constant across all pairs of values, then there is a constant of variation, indicating a direct variation. If the ratios are not constant, then the relationship does not exhibit a simple direct variation.
Let's break it down:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
X & 10 & 15 & 20 \\
\hline
Y & 6 & 4 & 3 \\
\hline
\end{array}
\][/tex]
### Step 1: Calculate the Ratios
- For [tex]\( X = 10 \)[/tex] and [tex]\( Y = 6 \)[/tex]:
[tex]\[
\frac{X}{Y} = \frac{10}{6} = 1.6667
\][/tex]
- For [tex]\( X = 15 \)[/tex] and [tex]\( Y = 4 \)[/tex]:
[tex]\[
\frac{15}{4} = 3.75
\][/tex]
- For [tex]\( X = 20 \)[/tex] and [tex]\( Y = 3 \)[/tex]:
[tex]\[
\frac{20}{3} = 6.6667
\][/tex]
### Step 2: Compare the Ratios
The calculated ratios are:
- [tex]\( \frac{10}{6} = 1.6667 \)[/tex]
- [tex]\( \frac{15}{4} = 3.75 \)[/tex]
- [tex]\( \frac{20}{3} = 6.6667 \)[/tex]
### Step 3: Analyze the Variation
Since the ratios between [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are not consistent (i.e., they are not equal to each other), we can conclude that there is no constant ratio of variation. This indicates that the relationship between [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] does not follow a simple direct or inverse variation.
To summarize:
- The ratios [tex]\([1.6667, 3.75, 6.6667]\)[/tex] are not the same.
- Therefore, there is no constant of variation between [tex]\( X \)[/tex] and [tex]\( Y \)[/tex].
