Finding the Value of x for Numbers in Proportion

The problem states that when a number, let's call it \(x\), is added to each of the numbers 11, 18, 27, and 42, the resulting numbers are in proportion.

Numbers in proportion mean that the ratio of the first two numbers is equal to the ratio of the last two numbers. So, if the numbers are \(a, b, c, d\), they are in proportion if \(\frac{a}{b} = \frac{c}{d}\).

After adding \(x\) to each number, the new numbers are:

• \(11 + x\)
• \(18 + x\)
• \(27 + x\)
• \(42 + x\)

Since these numbers are in proportion, we can write the equation:

\(\frac{11 + x}{18 + x} = \frac{27 + x}{42 + x}\)

To solve for \(x\), we can cross-multiply:

\((11 + x)(42 + x) = (27 + x)(18 + x)\)

Now, we expand both sides of the equation:

\(11 \times 42 + 11x + 42x + x^2 = 27 \times 18 + 27x + 18x + x^2\)

\(462 + 53x + x^2 = 486 + 45x + x^2\)

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

\(462 + 53x = 486 + 45x\)

Now, we collect the \(x\) terms on one side and the constant terms on the other side:

\(53x - 45x = 486 - 462\)

\(8x = 24\)

Finally, we solve for \(x\):

\(x = \frac{24}{8}\)

\(x = 3\)

So, the value of \(x\) is 3.

Calculating the Mean Proportional

The second part of the question asks for the mean proportional between \((11x + 3)\) and \((9x - 2)\).

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

First, we need to find the values of the two expressions using the value of \(x=3\) we just found:

Expression 1: \(11x + 3\)

Substitute \(x=3\): \(11(3) + 3 = 33 + 3 = 36\)

Expression 2: \(9x - 2\)

Substitute \(x=3\): \(9(3) - 2 = 27 - 2 = 25\)

Now, we need to find the mean proportional between 36 and 25.

Mean Proportional \( = \sqrt{36 \times 25}\)

Mean Proportional \( = \sqrt{900}\)

The square root of 900 is 30 because \(30 \times 30 = 900\).

Mean Proportional \( = 30\)

Thus, the mean proportional between \((11x + 3)\) and \((9x - 2)\) is 30.

Summary of Steps

1. Set up the proportion equation using the given numbers and \(x\).
2. Solve the equation for \(x\).
3. Substitute the value of \(x\) into the two given expressions.
4. Calculate the mean proportional between the values of the expressions.

Revision Table: Key Concepts

Additional Information: Solving Algebraic Equations

Solving algebraic equations involves finding the value(s) of the variable that make the equation true. In this problem, we solved a linear equation after expanding and simplifying the proportion.

Steps often involved:

• Simplify both sides of the equation (remove parentheses, combine like terms).
• Move all terms containing the variable to one side of the equation and constant terms to the other side.
• Combine like terms again.
• Isolate the variable by dividing or multiplying by the coefficient of the variable.

In our case, after expansion and cancellation of \(x^2\), we had \(462 + 53x = 486 + 45x\). We moved \(45x\) to the left (\(53x - 45x = 8x\)) and 462 to the right (\(486 - 462 = 24\)), resulting in \(8x = 24\), which was easily solved.