To solve this problem, we start by understanding the relationship described: [tex]\(15y\)[/tex] is inversely proportional to [tex]\(x\)[/tex]. Inverse proportionality means that as one variable increases, the other decreases such that their product remains constant.
Given the initial conditions:
- [tex]\( y = 20 \)[/tex] when [tex]\( x = 3 \)[/tex]
1. Identify the constant [tex]\(k\)[/tex]:
Since [tex]\(15y\)[/tex] is inversely proportional to [tex]\(x\)[/tex], we can write the relationship as:
[tex]\[
15y \cdot x = k
\][/tex]
where [tex]\(k\)[/tex] is a constant.
2. Calculate the constant [tex]\(k\)[/tex] using the initial conditions:
Using [tex]\( y = 20 \)[/tex] and [tex]\( x = 3 \)[/tex], we substitute these values into the equation:
[tex]\[
15 \cdot 20 \cdot 3 = k
\][/tex]
Simplifying, we get:
[tex]\[
k = 15 \cdot 20 \cdot 3 = 900
\][/tex]
3. Find the value of [tex]\(y\)[/tex] when [tex]\(x = 12\)[/tex]:
Using the constant [tex]\(k\)[/tex], we now need to find the new value of [tex]\(y\)[/tex] when [tex]\( x = 12 \)[/tex].
Rewriting our proportional relationship with the new [tex]\(x\)[/tex] value:
[tex]\[
15y \cdot 12 = k
\][/tex]
Substituting [tex]\(k = 900\)[/tex]:
[tex]\[
15y \cdot 12 = 900
\][/tex]
To find [tex]\(y\)[/tex], isolate it by dividing both sides of the equation by 15:
[tex]\[
y \cdot 12 = \frac{900}{15}
\][/tex]
Simplify the right-hand side:
[tex]\[
y \cdot 12 = 60
\][/tex]
Now, solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{60}{12} = 5
\][/tex]
Thus, the value of [tex]\(y\)[/tex] when [tex]\(x = 12\)[/tex] is [tex]\(\boxed{5}\)[/tex].
