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Identify the variation in the following examples:

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$X$[/tex] & 10 & 15 & 20 \\
\hline
[tex]$Y$[/tex] & 6 & 4 & 3 \\
\hline
\end{tabular}

Sagot :

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].