Understanding the Problem: Proportion and Mean Proportional

The question asks us to find a value, let's call it \(x\), which when added to four specific numbers (10, 16, 22, and 32) makes the new sequence of numbers proportional. This means the ratio of the first two numbers in the new sequence is equal to the ratio of the last two numbers. After finding the value of \(x\), we need to calculate the mean proportional between two expressions involving \(x\), specifically \((x + 1)\) and \((3x + 1)\).

Breaking Down the Steps

To solve this problem, we need to follow these steps:

• Set up an equation based on the proportion condition.
• Solve the equation to find the value of \(x\).
• Substitute the value of \(x\) into the expressions \((x + 1)\) and \((3x + 1)\).
• Calculate the mean proportional between the results from the previous step.

Solving for x in the Proportion

When \(x\) is added to each of the numbers 10, 16, 22, and 32, the new numbers are \(10+x\), \(16+x\), \(22+x\), and \(32+x\). If these numbers are in proportion, it means:

\(\frac{10+x}{16+x} = \frac{22+x}{32+x}\)

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

\((10+x)(32+x) = (16+x)(22+x)\)

Now, we expand both sides of the equation:

\(10 \times 32 + 10 \times x + x \times 32 + x \times x = 16 \times 22 + 16 \times x + x \times 22 + x \times x\)

\(320 + 10x + 32x + x^2 = 352 + 16x + 22x + x^2\)

Combine the \(x\) terms on each side:

\(320 + 42x + x^2 = 352 + 38x + x^2\)

Subtract \(x^2\) from both sides:

\(320 + 42x = 352 + 38x\)

Now, isolate the \(x\) terms on one side. Subtract \(38x\) from both sides:

\(320 + 42x - 38x = 352\)

\(320 + 4x = 352\)

Next, isolate the term with \(x\). Subtract 320 from both sides:

\(4x = 352 - 320\)

\(4x = 32\)

Finally, solve for \(x\) by dividing by 4:

\(x = \frac{32}{4}\)

\(x = 8\)

So, the value of \(x\) is 8.

Calculating the Mean Proportional

The question asks for the mean proportional between \((x + 1)\) and \((3x + 1)\). We found that \(x = 8\).

First, calculate the values of the two numbers:

• First number: \(x + 1 = 8 + 1 = 9\)
• Second number: \(3x + 1 = 3(8) + 1 = 24 + 1 = 25\)

The mean proportional between two numbers, say \(a\) and \(b\), is given by the formula \(\sqrt{a \times b}\). In this case, \(a = 9\) and \(b = 25\).

Mean Proportional = \(\sqrt{9 \times 25}\)

Mean Proportional = \(\sqrt{225}\)

The square root of 225 is 15.

Mean Proportional = 15

Verification

Let's verify if adding \(x=8\) to the original numbers puts them in proportion:

• \(10 + 8 = 18\)
• \(16 + 8 = 24\)
• \(22 + 8 = 30\)
• \(32 + 8 = 40\)

The new sequence is 18, 24, 30, 40. Let's check the ratios:

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

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

Since \(\frac{18}{24} = \frac{30}{40}\), the numbers are indeed in proportion when \(x=8\). Our calculation for \(x\) is correct.

Revision Table: Key Concepts

Additional Information: Ratios and Proportionality

A ratio is a comparison of two quantities. For example, the ratio of 18 to 24 is \(\frac{18}{24}\) or 18:24. This ratio can be simplified to \(\frac{3}{4}\) or 3:4.

When four numbers \(a, b, c, d\) are in proportion, it means that the ratio of the first two is equal to the ratio of the last two. This is written as \(a:b :: c:d\), which is equivalent to \(\frac{a}{b} = \frac{c}{d}\). In a proportion \(a:b :: c:d\), \(a\) and \(d\) are called the 'extremes', and \(b\) and \(c\) are called the 'means'. A key property of proportion is that the product of the extremes is equal to the product of the means, i.e., \(a \times d = b \times c\). We used this property (cross-multiplication) to solve for \(x\) in our problem.

The mean proportional (also called the geometric mean) between two positive numbers \(a\) and \(b\) is a number \(m\) such that \(a:m :: m:b\), which means \(\frac{a}{m} = \frac{m}{b}\). Cross-multiplying gives \(m^2 = ab\), so \(m = \sqrt{ab}\). This is the formula we used to find the mean proportional between \((x+1)\) and \((3x+1)\).