Understanding and Solving the Proportion Problem

This problem requires us to understand the concepts of fourth proportion and third proportion and then use the given condition to find the value of 'k'.

Calculating the Fourth Proportion

When four numbers a, b, c, and d are in proportion, it means that the ratio of the first two is equal to the ratio of the last two. This is written as a : b :: c : d, which is equivalent to $\frac{a}{b} = \frac{c}{d}$. In this case, 'd' is called the fourth proportion.

We are given the numbers 12, 18, and 6, and we need to find the fourth proportion. Let the fourth proportion be 'x'. So, we have the proportion:

$12 : 18 :: 6 : x$

Writing this as an equation:

$\frac{12}{18} = \frac{6}{x}$

To solve for x, we can cross-multiply:

$12 \times x = 18 \times 6$

$12x = 108$

Now, divide by 12 to find x:

$x = \frac{108}{12}$

$x = 9$

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

Calculating the Third Proportion

When three numbers a, b, and c are in proportion, and 'b' is the mean proportion, it means that a : b :: b : c, which is equivalent to $\frac{a}{b} = \frac{b}{c}$. In this case, 'c' is called the third proportion to a and b.

We are given the numbers k and 6, and 6 is the mean proportion. We need to find the third proportion. Let the third proportion be 'y'. So, we have the proportion:

$k : 6 :: 6 : y$

Writing this as an equation:

$\frac{k}{6} = \frac{6}{y}$

Equating the Proportions and Solving for k

The problem states that the fourth proportion (which we found to be 9) is the same as the third proportion to k and 6. This means the value of 'y' from the second step is 9.

So, we substitute y = 9 into the equation for the third proportion:

$\frac{k}{6} = \frac{6}{9}$

Now, we need to solve this equation for k. First, we can simplify the fraction on the right side:

$\frac{6}{9} = \frac{2 \times 3}{3 \times 3} = \frac{2}{3}$

So the equation becomes:

$\frac{k}{6} = \frac{2}{3}$

To find k, multiply both sides by 6:

$k = \frac{2}{3} \times 6$

$k = \frac{12}{3}$

$k = 4$

The value of k is 4.

Conclusion

The fourth proportion to 12, 18, and 6 is 9. The third proportion to k and 6 is also 9. By setting up the equation $\frac{k}{6} = \frac{6}{9}$ and solving for k, we find that $k=4$.

Revision Table: Key Proportion Concepts

Additional Information: Types of Proportion

Beyond basic proportions, there are different types based on how quantities relate:

• Direct Proportion: If two quantities increase or decrease together at a constant ratio. Example: Cost of items increases as the number of items increases (at a fixed price per item).
• Inverse Proportion: If one quantity increases as the other decreases, such that their product is constant. Example: Speed and time taken to cover a fixed distance. As speed increases, time decreases.

Understanding these relationships helps in solving various problems involving ratios and proportions.