To find the constant of variation [tex]\( k \)[/tex] in the equation where [tex]\( j \)[/tex] varies directly with [tex]\( m \)[/tex] and inversely with [tex]\( p \)[/tex], we start with the direct variation equation:
[tex]\[
j = k \frac{m}{p}
\][/tex]
Given values:
- [tex]\( m = 2 \)[/tex]
- [tex]\( p = \frac{1}{4} \)[/tex]
- [tex]\( j = 32 \)[/tex]
We need to solve for [tex]\( k \)[/tex]. First, substitute the given values into the equation:
[tex]\[
32 = k \frac{2}{\frac{1}{4}}
\][/tex]
Simplify the fraction [tex]\(\frac{2}{\frac{1}{4}}\)[/tex]:
[tex]\[
\frac{2}{\frac{1}{4}} = 2 \times 4 = 8
\][/tex]
This simplifies our equation to:
[tex]\[
32 = k \times 8
\][/tex]
Next, solve for [tex]\( k \)[/tex] by dividing both sides of the equation by 8:
[tex]\[
k = \frac{32}{8} = 4
\][/tex]
Therefore, the constant of variation is [tex]\( \boxed{4} \)[/tex].
