Given that [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex], it means that the product of [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is a constant. Mathematically, we express this relationship as:
[tex]\[ y \cdot x = k \][/tex]
where [tex]\( k \)[/tex] is a constant.
We are given that [tex]\( y = 20 \)[/tex] when [tex]\( x = 3 \)[/tex]. We substitute these values into the equation to find the value of [tex]\( k \)[/tex]:
[tex]\[ 20 \cdot 3 = k \][/tex]
[tex]\[ k = 60 \][/tex]
Now that we have established the constant [tex]\( k \)[/tex] as 60, we need to find the value of [tex]\( y \)[/tex] when [tex]\( x = 12 \)[/tex].
Using the relationship [tex]\( y \cdot x = k \)[/tex], we substitute [tex]\( k = 60 \)[/tex] and [tex]\( x = 12 \)[/tex]:
[tex]\[ y \cdot 12 = 60 \][/tex]
To solve for [tex]\( y \)[/tex], we divide both sides of the equation by 12:
[tex]\[ y = \frac{60}{12} \][/tex]
[tex]\[ y = 5 \][/tex]
Thus, the value of [tex]\( y \)[/tex] when [tex]\( x = 12 \)[/tex] is 5.
