Understanding Mean Proportional

The mean proportional of two positive numbers, say 'a' and 'b', is a number 'x' such that the ratio of 'a' to 'x' is equal to the ratio of 'x' to 'b'. This relationship can be written as a proportion:

\(\frac{a}{x} = \frac{x}{b}\)

From this proportion, we can find the formula for the mean proportional 'x'. By cross-multiplying, we get:

\(x^2 = a \times b\)

Taking the square root of both sides gives us the formula:

\(x = \sqrt{a \times b}\)

So, the mean proportional of two numbers is the square root of their product. It is also known as the geometric mean of the two numbers.

Calculating the Mean Proportional of 40 and 90

We need to find the mean proportional of the numbers 40 and 90. Using the formula \(x = \sqrt{a \times b}\), where \(a = 40\) and \(b = 90\), we can substitute these values into the formula.

Step 1: Identify the two numbers.

The given numbers are 40 and 90.

Step 2: Write down the formula for the mean proportional.

\(\text{Mean Proportional} = \sqrt{\text{Number 1} \times \text{Number 2}}\)

Step 3: Substitute the numbers into the formula.

\(\text{Mean Proportional} = \sqrt{40 \times 90}\)

Step 4: Calculate the product of the two numbers.

\(40 \times 90 = 3600\)

So, the expression becomes:

\(\text{Mean Proportional} = \sqrt{3600}\)

Step 5: Find the square root of the product.

To find the square root of 3600, we can think of it as \(36 \times 100\). The square root of \(36\) is \(6\), and the square root of \(100\) is \(10\).

\(\sqrt{3600} = \sqrt{36 \times 100} = \sqrt{36} \times \sqrt{100} = 6 \times 10 = 60\)

Alternatively, you can use prime factorization or a calculator to find the square root of 3600, which is 60.

Therefore, the mean proportional of 40 and 90 is 60.

Verification

To verify, check if 40, 60, and 90 form a proportion where 60 is the mean term:

\(\frac{40}{60} = \frac{2}{3}\)

\(\frac{60}{90} = \frac{6 \times 10}{9 \times 10} = \frac{6}{9} = \frac{2}{3}\)

Since \(\frac{40}{60} = \frac{60}{90} = \frac{2}{3}\), 60 is indeed the mean proportional of 40 and 90.

The result is 60.