Finding the Difference Between Two Numbers

Let the two numbers be \(a\) and \(b\). We are given two pieces of information about these numbers:

1. The average of the two numbers is 13.
2. The square root of their product is 12.

From the definition of average, we can write the first piece of information as an equation:

\(\frac{a+b}{2} = 13\)

Multiplying both sides by 2, we get the sum of the two numbers:

\(a+b = 13 \times 2\)

\(a+b = 26\)

From the definition of the square root of the product, we can write the second piece of information as an equation:

\(\sqrt{ab} = 12\)

Squaring both sides to find the product of the two numbers:

\(ab = 12^2\)

\(ab = 144\)

Now we have the sum (\(a+b=26\)) and the product (\(ab=144\)) of the two numbers. We need to find the difference between the numbers, which is \(|a-b|\).

We can use a common algebraic identity that relates the square of the difference to the square of the sum and the product:

\((a-b)^2 = (a+b)^2 - 4ab\)

Substitute the values we found for \(a+b\) and \(ab\) into this identity:

\((a-b)^2 = (26)^2 - 4 \times 144\)

Calculate the values:

\((a-b)^2 = 676 - 576\)

\((a-b)^2 = 100\)

To find the difference \(a-b\), take the square root of both sides:

\(a-b = \pm \sqrt{100}\)

\(a-b = \pm 10\)

The difference between the numbers can be 10 or -10, depending on which number is considered first. However, typically when asked for "the difference", we provide the positive value, which represents the distance between the two numbers on the number line. Therefore, the difference is 10.

Revision Table: Average, Product, and Difference

Additional Information: Relation Between AM, GM, and HM

This problem involves the relationship between two numbers through their average and product. In mathematics, the average is also known as the Arithmetic Mean (AM), and the square root of the product of two positive numbers is their Geometric Mean (GM).

• Arithmetic Mean (AM): For two numbers \(a\) and \(b\), AM = \(\frac{a+b}{2}\).
• Geometric Mean (GM): For two positive numbers \(a\) and \(b\), GM = \(\sqrt{ab}\).
• Harmonic Mean (HM): For two numbers \(a\) and \(b\), HM = \(\frac{2}{\frac{1}{a}+\frac{1}{b}} = \frac{2ab}{a+b}\).

There is a relationship between AM, GM, and HM. For positive numbers, AM \(\ge\) GM \(\ge\) HM. Also, for two numbers, \(GM^2 = AM \times HM\).

In this problem:

• AM = 13
• GM = 12

Since \(13 \ge 12\), the AM \(\ge\) GM property holds true.

We could also find the Harmonic Mean using the relationship \(GM^2 = AM \times HM\):

\(12^2 = 13 \times HM\)

\(144 = 13 \times HM\)

\(HM = \frac{144}{13}\)

Note that \(\frac{144}{13} \approx 11.08\), and \(13 \ge 12 \ge 11.08\), confirming the AM \(\ge\) GM \(\ge\) HM relationship.