Fourth proportion to 12, 18, and 6 is same as the third proportion to k and 6. What is the value of k?

First recall the standard definitions:
• If $$a:b = c:d$$, then $$d$$ is called the fourth proportion to $$a,\,b,\,c$$.
• If $$a:b = b:c$$, then $$c$$ is called the third proportion to $$a,\,b$$.

Step 1: Find the fourth proportion to 12, 18, 6.
Let this fourth proportion be $$x$$. By definition
$$\frac{12}{18} = \frac{6}{x}$$

Cross-multiply:
$$12x = 18 \times 6$$
$$x = \frac{18 \times 6}{12} = 9$$

So the fourth proportion is $$9$$.

Step 2: Express the third proportion to $$k$$ and $$6$$.
Let this third proportion be $$y$$. By definition
$$\frac{k}{6} = \frac{6}{y}$$

Cross-multiply:
$$ky = 36$$
$$y = \frac{36}{k}$$

Step 3: Equate the two proportions.
The problem states that the fourth proportion just found is the same as this third proportion, so
$$y = x$$
$$\frac{36}{k} = 9$$

Solve for $$k$$:
$$36 = 9k$$
$$k = \frac{36}{9} = 4$$

Therefore, $$k = 4$$.

Option B which is: 4