Understanding Proportion and Solving for x

The problem states that when a number, let's call it \(x\), is subtracted from each of 23, 39, 32, and 56, the resulting numbers are in proportion. This means that the ratio of the first two numbers is equal to the ratio of the last two numbers.

The numbers obtained after subtracting \(x\) are:

• \(23 - x\)
• \(39 - x\)
• \(32 - x\)
• \(56 - x\)

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

\(\frac{23 - x}{39 - x} = \frac{32 - x}{56 - x}\)

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

\((23 - x)(56 - x) = (32 - x)(39 - x)\)

Now, we expand both sides of the equation:

\(23 \times 56 - 23x - 56x + x^2 = 32 \times 39 - 32x - 39x + x^2\)

\(1288 - 79x + x^2 = 1248 - 71x + x^2\)

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

\(1288 - 79x = 1248 - 71x\)

Now, let's rearrange the terms to isolate \(x\). We can add \(79x\) to both sides and subtract \(1248\) from both sides:

\(1288 - 1248 = 79x - 71x\)

\(40 = 8x\)

Divide by 8 to find the value of \(x\):

\(x = \frac{40}{8}\)

\(x = 5\)

So, the value of \(x\) is 5.

Calculating the Mean Proportional

The question asks for the mean proportional between \((x + 4)\) and \((3x + 1)\). First, we need to find the values of these two expressions using the calculated value of \(x=5\).

The first term is \((x + 4)\):

\(x + 4 = 5 + 4 = 9\)

The second term is \((3x + 1)\):

\(3x + 1 = 3(5) + 1 = 15 + 1 = 16\)

The numbers are 9 and 16.

The mean proportional between two numbers \(a\) and \(b\) is given by the formula \(\sqrt{ab}\).

In this case, \(a = 9\) and \(b = 16\). The mean proportional is:

\(\sqrt{9 \times 16}\)

Calculate the product inside the square root:

\(\sqrt{144}\)

The square root of 144 is 12.

\(\sqrt{144} = 12\)

Therefore, the mean proportional between \((x + 4)\) and \((3x + 1)\) is 12.

Step-by-Step Solution Summary

1. Set up the proportion equation: \(\frac{23 - x}{39 - x} = \frac{32 - x}{56 - x}\).
2. Cross-multiply and expand: \((23 - x)(56 - x) = (32 - x)(39 - x) \Rightarrow 1288 - 79x + x^2 = 1248 - 71x + x^2\).
3. Solve for \(x\): \(1288 - 1248 = 79x - 71x \Rightarrow 40 = 8x \Rightarrow x = 5\).
4. Calculate the numbers for the mean proportional: \((x+4) = (5+4) = 9\) and \((3x+1) = (3(5)+1) = 16\).
5. Calculate the mean proportional: \(\sqrt{9 \times 16} = \sqrt{144} = 12\).

Revision Table: Proportion and Mean Proportional Concepts

Additional Information on Proportionality

When four numbers \(a, b, c, d\) are in proportion, written as \(a : b :: c : d\), it means that the ratio of the first term to the second term is equal to the ratio of the third term to the fourth term. Mathematically, this is expressed as \(\frac{a}{b} = \frac{c}{d}\).

In a proportion \(a : b :: c : d\):

• \(a\) and \(d\) are called the extreme terms.
• \(b\) and \(c\) are called the mean terms.

A key property of proportion is that the product of the extremes is equal to the product of the means:

\(a \times d = b \times c\)

This property was used in solving the problem when we performed cross-multiplication: \((23-x)(56-x) = (39-x)(32-x)\).

The concept of mean proportional is a special case of proportion. If three numbers \(a, b, c\) are in continued proportion, it means \(a : b :: b : c\), or \(\frac{a}{b} = \frac{b}{c}\). In this case, \(b\) is called the mean proportional between \(a\) and \(c\).