Understanding the Problem: Numbers in Proportion

The question states that when a certain value, let's call it 'x', is subtracted from four specific numbers (21, 22, 60, and 64), the resulting numbers, in that specific order, are in proportion. Our first goal is to find the value of this unknown 'x'. Once we have 'x', we need to calculate the mean proportional between two expressions involving 'x': (x + 1) and (7x + 8).

Setting up the Proportion Equation

When four numbers, say a, b, c, and d, are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. Mathematically, this is written as:

\(\frac{a}{b} = \frac{c}{d}\) or \(a:b = c:d\)

In this problem, the numbers after subtracting 'x' are (21-x), (22-x), (60-x), and (64-x). Since these are in proportion, we can write the equation:

\(\frac{21-x}{22-x} = \frac{60-x}{64-x}\)

Solving for the Value of x

To solve this equation for 'x', we can cross-multiply:

\((21-x)(64-x) = (22-x)(60-x)\)

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

\(21 \times 64 - 21x - 64x + (-x)(-x) = 22 \times 60 - 22x - 60x + (-x)(-x)\)

\(1344 - 85x + x^2 = 1320 - 82x + x^2\)

Notice that \(x^2\) appears on both sides of the equation. We can subtract \(x^2\) from both sides:

\(1344 - 85x = 1320 - 82x\)

Now, let's gather the 'x' terms on one side and the constant terms on the other. We can add \(85x\) to both sides and subtract \(1320\) from both sides:

\(1344 - 1320 = -82x + 85x\)

\(24 = 3x\)

Finally, divide by 3 to find the value of x:

\(x = \frac{24}{3}\)

\(x = 8\)

So, the value of 'x' is 8.

Calculating the Mean Proportional

The question asks for the mean proportional between (x + 1) and (7x + 8). The mean proportional between two numbers, say 'a' and 'b', is the square root of their product, i.e., \(\sqrt{a \times b}\).

First, let's find the values of the two expressions using \(x=8\):

• The first expression is (x + 1): \(8 + 1 = 9\)
• The second expression is (7x + 8): \(7 \times 8 + 8 = 56 + 8 = 64\)

Now, we find the mean proportional between 9 and 64:

Mean Proportional = \(\sqrt{(x+1)(7x+8)}\)

Mean Proportional = \(\sqrt{9 \times 64}\)

We know that \(\sqrt{9} = 3\) and \(\sqrt{64} = 8\). So,

Mean Proportional = \(3 \times 8\)

Mean Proportional = \(24\)

The mean proportional between (x + 1) and (7x + 8) is 24.

Summary of Steps

Let's quickly recap the steps:

1. Identify the numbers and the operation (subtracting x).
2. Set up the proportion based on the resulting numbers.
3. Solve the equation for x by cross-multiplication and simplification.
4. Calculate the values of the expressions (x+1) and (7x+8) using the found value of x.
5. Calculate the mean proportional of these two resulting numbers.

Revision Table: Key Concepts Revisited

Additional Information: Mean Proportional and Geometric Mean

The mean proportional is also known as the geometric mean for two numbers. For a set of 'n' numbers, the geometric mean is the n-th root of their product. For two numbers (n=2), the geometric mean is the square root of their product, which is exactly the mean proportional.

The concept of proportion is fundamental in various mathematical and real-world applications, including scaling drawings, maps, mixing ratios, and understanding relationships between quantities.

Solving algebraic equations, as we did to find 'x', is a crucial skill in mathematics. It involves isolating the variable using inverse operations while maintaining the equality of the equation.