Understanding the Problem: Proportion and Mean Proportional

The question asks us to find the mean proportional between two expressions, (3x - 2) and (5x + 4), where 'x' is a value that makes the numbers 9, 15, 21, and 31 proportional when added to each of them.

When four numbers, say 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. Mathematically, this is expressed as \(\frac{a}{b} = \frac{c}{d}\). In this problem, the numbers obtained after adding 'x' are (9+x), (15+x), (21+x), and (31+x). Therefore, they are in proportion if:

\[ \frac{9+x}{15+x} = \frac{21+x}{31+x} \]

The mean proportional between two numbers, 'a' and 'b', is the square root of their product, i.e., \(\sqrt{ab}\).

Finding the Value of x

We need to solve the proportion equation to find the value of x.

The equation is:

\[ \frac{9+x}{15+x} = \frac{21+x}{31+x} \]

Cross-multiplying gives:

\[ (9+x)(31+x) = (15+x)(21+x) \]

Expand both sides of the equation:

\[ 9 \times 31 + 9 \times x + x \times 31 + x \times x = 15 \times 21 + 15 \times x + x \times 21 + x \times x \]

\[ 279 + 9x + 31x + x^2 = 315 + 15x + 21x + x^2 \]

Combine like terms on each side:

\[ 279 + 40x + x^2 = 315 + 36x + x^2 \]

Subtract \(x^2\) from both sides of the equation:

\[ 279 + 40x = 315 + 36x \]

Now, isolate the terms with x on one side and constant terms on the other. Subtract 36x from both sides:

\[ 279 + 40x - 36x = 315 \]

\[ 279 + 4x = 315 \]

Subtract 279 from both sides:

\[ 4x = 315 - 279 \]

\[ 4x = 36 \]

Divide by 4 to find x:

\[ x = \frac{36}{4} \]

\[ x = 9 \]

Calculating the Numbers for Mean Proportional

Now that we have found \(x = 9\), we can find the two numbers for which we need to calculate the mean proportional. The numbers are given by (3x - 2) and (5x + 4).

First number:

\[ 3x - 2 = 3(9) - 2 = 27 - 2 = 25 \]

Second number:

\[ 5x + 4 = 5(9) + 4 = 45 + 4 = 49 \]

The two numbers are 25 and 49.

Finding the Mean Proportional

The mean proportional between 25 and 49 is \(\sqrt{25 \times 49}\).

\[ \text{Mean Proportional} = \sqrt{25 \times 49} \]

Since \(25 = 5^2\) and \(49 = 7^2\), we have:

\[ \text{Mean Proportional} = \sqrt{5^2 \times 7^2} \]

Using the property \(\sqrt{a^2 b^2} = \sqrt{(ab)^2} = ab\):

\[ \text{Mean Proportional} = \sqrt{(5 \times 7)^2} = 5 \times 7 = 35 \]

The mean proportional between (3x - 2) and (5x + 4) when \(x=9\) is 35.

Revision Table: Key Concepts

Additional Information: Types of Proportion

Besides the type of proportion discussed in the problem (four numbers in proportion), there are other related concepts:

• Continuous Proportion: Three numbers a, b, c are in continuous proportion if \(a/b = b/c\). Here, 'b' is called the mean proportional between 'a' and 'c'.
• Direct Proportion: Two quantities are in direct proportion if an increase in one quantity causes a proportional increase in the other, and vice versa. The ratio of the two quantities is constant. Example: Distance covered is directly proportional to time at a constant speed.
• Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity causes a proportional decrease in the other, and vice versa. The product of the two quantities is constant. Example: Time taken to complete a work is inversely proportional to the number of workers.

Understanding these types of proportions helps in solving various problems involving ratios and proportionality.