Understanding Proportion Concepts

This problem involves understanding and applying the concepts of fourth proportion and third proportion. We need to find the value of 'k' by first calculating the fourth proportion of three given numbers and then using that value as the third proportion in a different set of numbers.

Calculating the Fourth Proportion

The question asks for the fourth proportion to 12, 18, and 6. If four quantities $a, b, c,$ and $d$ are in proportion, they are written as $a:b::c:d$, which means $\frac{a}{b} = \frac{c}{d}$. Here, $a=12$, $b=18$, $c=6$, and we need to find the fourth proportion, let's call it $x$.

So, the proportion is $12:18::6:x$.

This can be written as a fraction:

\(\frac{12}{18} = \frac{6}{x}\)

To solve for $x$, we can cross-multiply:

\(12 \times x = 18 \times 6\)

\(12x = 108\)

Now, divide both sides by 12:

\(x = \frac{108}{12}\)

\(x = 9\)

So, the fourth proportion to 12, 18, 6 is 9.

Calculating the Third Proportion

The question states that this fourth proportion (which we found to be 9) is equal to the third proportion to 4 and k. If three quantities $a, b,$ and $c$ are in continued proportion, they are written as $a:b::b:c$, which means $\frac{a}{b} = \frac{b}{c}$. Here, $a=4$, $b=k$, and the third proportion $c$ is equal to the fourth proportion we calculated, which is 9.

So, the proportion is $4:k::k:9$.

This can be written as a fraction:

\(\frac{4}{k} = \frac{k}{9}\)

To solve for $k$, we can cross-multiply:

\(4 \times 9 = k \times k\)

\(36 = k^2\)

To find $k$, we take the square root of both sides:

\(k = \sqrt{36}\)

\(k = \pm 6\)

In the context of proportion problems involving lengths or positive quantities (which is typical unless stated otherwise), we usually consider the positive value. Therefore, $k=6$.

Step-by-Step Solution Summary

1. Identify the numbers for the fourth proportion calculation: 12, 18, 6.
2. Set up the proportion: $\frac{12}{18} = \frac{6}{x}$.
3. Solve for $x$ (the fourth proportion): $12x = 108 \implies x = 9$.
4. Identify the numbers for the third proportion calculation: 4, k. The third proportion is 9.
5. Set up the continued proportion: $\frac{4}{k} = \frac{k}{9}$.
6. Solve for $k$: $k^2 = 36 \implies k = 6$ (considering the positive value).

The value of k is 6.

Revision Table: Key Proportion Concepts

Additional Information on Ratio and Proportion

Ratio and proportion are fundamental concepts in mathematics used to compare quantities and express relationships between them. A ratio is a comparison of two quantities by division, often written as $a:b$ or $\frac{a}{b}$. A proportion is an equation stating that two ratios are equal, e.g., $\frac{a}{b} = \frac{c}{d}$.

In a proportion $a:b::c:d$, the terms $a$ and $d$ are called the 'extremes', and $b$ and $c$ are called the 'means'. A key property of proportion is that the product of the means is equal to the product of the extremes, i.e., $b \times c = a \times d$. This property is often used to solve for an unknown term in a proportion.

Continued proportion is a special case where the middle terms are the same, as seen in the definition of third proportion and mean proportion. Understanding these different types of proportions is crucial for solving related problems in various mathematical contexts.