Understanding the Problem: Proportion and Mean Proportional

The question asks us to find a value, let's call it \(x\), such that when we subtract \(x\) from four specific numbers (24, 30, 36, and 46), the resulting new numbers form a proportion. This means the ratio of the first two new numbers is equal to the ratio of the last two new numbers. After finding \(x\), we need to calculate the mean proportional between two expressions that depend on \(x\): \( (2x + 3) \) and \( (3x + 2) \).

Setting up the Proportion

When \(x\) is subtracted from each number, the new numbers are:

• \( 24 - x \)
• \( 30 - x \)
• \( 36 - x \)
• \( 46 - x \)

Since these numbers are in proportion in this order, we can write the relationship as:

\( \frac{24 - x}{30 - x} = \frac{36 - x}{46 - x} \)

Solving for the Value of x

To find the value of \(x\), we need to solve this equation. We can do this by cross-multiplication:

\( (24 - x)(46 - x) = (30 - x)(36 - x) \)

Now, let's expand both sides of the equation:

Left side: \( 24 \times 46 - 24x - 46x + x^2 = 1104 - 70x + x^2 \)

Right side: \( 30 \times 36 - 30x - 36x + x^2 = 1080 - 66x + x^2 \)

So, the equation becomes:

\( 1104 - 70x + x^2 = 1080 - 66x + x^2 \)

We can subtract \(x^2\) from both sides:

\( 1104 - 70x = 1080 - 66x \)

Now, let's gather the \(x\) terms on one side and the constant terms on the other:

\( 1104 - 1080 = 70x - 66x \)

\( 24 = 4x \)

Finally, divide by 4 to find \(x\):

\( x = \frac{24}{4} \)

\( x = 6 \)

Let's quickly check if \(x=6\) makes the numbers proportional:

• \( 24 - 6 = 18 \)
• \( 30 - 6 = 24 \)
• \( 36 - 6 = 30 \)
• \( 46 - 6 = 40 \)

The proportion is \( \frac{18}{24} = \frac{30}{40} \). Simplifying the fractions:

\( \frac{18}{24} = \frac{3 \times 6}{4 \times 6} = \frac{3}{4} \)

\( \frac{30}{40} = \frac{3 \times 10}{4 \times 10} = \frac{3}{4} \)

Since \( \frac{3}{4} = \frac{3}{4} \), the numbers are indeed in proportion when \(x=6\).

Calculating the Expressions (2x + 3) and (3x + 2)

Now that we know \(x = 6\), we can find the values of the two expressions:

First expression: \( 2x + 3 = 2(6) + 3 = 12 + 3 = 15 \)

Second expression: \( 3x + 2 = 3(6) + 2 = 18 + 2 = 20 \)

We need to find the mean proportional between 15 and 20.

Finding the Mean Proportional

The mean proportional between two numbers, say \(a\) and \(b\), is given by the formula \( \sqrt{a \times b} \). In this case, \(a = 15\) and \(b = 20\).

Mean Proportional \( = \sqrt{15 \times 20} \)

Let's calculate the product and find the square root:

\( \sqrt{15 \times 20} = \sqrt{300} \)

To simplify \( \sqrt{300} \), we can look for perfect square factors of 300. We know that \( 300 = 100 \times 3 \), and 100 is a perfect square (\( 10^2 \)).

\( \sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10 \times \sqrt{3} \)

So, the mean proportional is \( 10\sqrt{3} \).

Final Answer Derivation

We found that the mean proportional between \( (2x+3) \) and \( (3x+2) \) when \(x=6\) is \( 10\sqrt{3} \).

Revision Table: Proportion and Mean Proportional

Additional Information: Solving Proportionality Problems

Proportionality problems often involve setting up algebraic equations and solving for an unknown variable. When four numbers \(a, b, c, d\) are in proportion, it means \(a:b :: c:d\), which is equivalent to \( \frac{a}{b} = \frac{c}{d} \). Cross-multiplication is a standard technique to solve such equations: \( ad = bc \).

The concept of mean proportional is a special case of continued proportion. If \(a, m, b\) are in continued proportion, then \( \frac{a}{m} = \frac{m}{b} \), where \(m\) is the mean proportional. This leads to \(m^2 = ab\).

When solving for \(x\) in proportionality problems, it's a good practice to substitute the value of \(x\) back into the original expressions to verify that the proportion holds. This helps catch any calculation errors.

Remember to simplify square roots when calculating the mean proportional by factoring out perfect squares.