If X equals 40 when Y equals 20, what is the constant of variation for the variables X and Y that vary directly?

When two variables vary directly, it means that their relationship can be expressed as (X = kY), where (k) is the constant of variation. In this case, we are given that (X = 40) when (Y = 20).

To find the constant of variation, we can substitute the known values into the equation:

[

40 = k \cdot 20

]

To solve for (k), we divide both sides by 20:

[

k = \frac{40}{20}

]

Calculating this gives:

[

k = 2

]

This indicates that for every unit increase in (Y), (X) increases by a factor of 2. The constant of variation is therefore 2, which corresponds to the option provided.

This demonstrates a consistent proportional relationship between (X) and (Y) in which (X) is always double the value of (Y) when the relationship holds. This understanding helps in solving similar problems involving direct variation.