To determine which of the given options correctly expresses [tex]\( k \)[/tex] in terms of [tex]\( j \)[/tex] when [tex]\(\frac{j}{k} = \frac{4}{5}\)[/tex], follow these steps:
1. We start with the given equation:
[tex]\[
\frac{j}{k} = \frac{4}{5}
\][/tex]
2. To isolate [tex]\( k \)[/tex], multiply both sides of the equation by [tex]\( k \)[/tex]:
[tex]\[
j = \frac{4}{5}k
\][/tex]
3. Now, to solve for [tex]\( k \)[/tex], we need to get [tex]\( k \)[/tex] by itself. We do this by dividing both sides by [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
k = \frac{j}{\frac{4}{5}}
\][/tex]
4. Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of [tex]\(\frac{4}{5}\)[/tex] is [tex]\(\frac{5}{4}\)[/tex]:
[tex]\[
k = j \times \frac{5}{4}
\][/tex]
5. Therefore, [tex]\( k \)[/tex] in terms of [tex]\( j \)[/tex] is:
[tex]\[
k = \frac{5j}{4}
\][/tex]
Given the options:
- (A) [tex]\( k = \frac{4}{5 j} \)[/tex]
- (B) [tex]\( k = \frac{5}{4 j} \)[/tex]
- (C) [tex]\( k = \frac{4 j}{5} \)[/tex]
- (D) [tex]\( k = \frac{5 j}{4} \)[/tex]
The correct answer is [tex]\( k = \frac{5 j}{4} \)[/tex], which corresponds to option (D).
