To solve for [tex]\( x \)[/tex] using a proportion, you need to find the proportion where [tex]\( x \)[/tex] is isolated in a way that allows for a straightforward calculation. Let's consider each of the given options:
### Option a:
[tex]\[
\frac{48}{36} = \frac{x}{48}
\][/tex]
This ratio does not isolate [tex]\( x \)[/tex] directly in one of the proportions. We would need to perform extra steps to solve for [tex]\( x \)[/tex].
### Option b:
[tex]\[
\frac{x}{36} = \frac{36}{48}
\][/tex]
In this proportion, [tex]\( x \)[/tex] is alone in the numerator of the first ratio. This makes it easier to solve for [tex]\( x \)[/tex] using cross-multiplication:
[tex]\[
x \cdot 48 = 36 \cdot 36
\][/tex]
Then solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{36 \times 36}{48}
\][/tex]
### Option c:
[tex]\[
\frac{x}{36} = \frac{48}{x}
\][/tex]
While this proportion is valid, solving it algebraically would involve more steps, including solving a quadratic equation. This is not an immediate direct solution.
### Option d:
[tex]\[
\frac{x}{84} = \frac{36}{48}
\][/tex]
Similar to the others, this proportion also isolates [tex]\( x \)[/tex] but requires additional steps to solve. This option can also cause confusion due to the 84 denominator.
### Conclusion:
Among all the given options, option b:
[tex]\[
\frac{x}{36} = \frac{36}{48}
is the simplest and most direct way to solve for \( x \). This proportion will yield the solution with the least number of steps.
Therefore, the correct choice is:
\[
\boxed{b}
\][/tex]
