Understanding Harmonic Mean and Solving for an Unknown

The question asks us to find the value of 'x' given that the harmonic mean of 60 and x is 48. The harmonic mean is a type of average that is calculated in a specific way, particularly useful in situations involving rates or ratios.

What is Harmonic Mean?

For two numbers, say 'a' and 'b', the harmonic mean (HM) is defined as the reciprocal of the arithmetic mean of the reciprocals of the numbers. In simpler terms, for two numbers, the formula is:

\[ \text{HM} = \frac{2}{\frac{1}{a} + \frac{1}{b}} \]

This formula can be rearranged to be more convenient for calculations:

\[ \text{HM} = \frac{2ab}{a+b} \]

Applying the Harmonic Mean Formula

We are given:

• One number (let's say 'a') = 60
• The other number (let's say 'b') = x
• The Harmonic Mean (HM) = 48

We can plug these values into the harmonic mean formula for two numbers:

\[ 48 = \frac{2 \times 60 \times x}{60 + x} \]

Solving for the Unknown Value 'x'

Now, we need to solve this equation to find the value of x. Let's simplify and rearrange the equation:

1. Multiply both sides by \((60 + x)\) to remove the denominator: \[ 48 \times (60 + x) = 2 \times 60 \times x \] \[ 48(60 + x) = 120x \]
2. Distribute 48 on the left side: \[ 48 \times 60 + 48 \times x = 120x \] \[ 2880 + 48x = 120x \]
3. Subtract \(48x\) from both sides to isolate the terms with x on one side: \[ 2880 = 120x - 48x \] \[ 2880 = (120 - 48)x \] \[ 2880 = 72x \]
4. Divide both sides by 72 to find the value of x: \[ x = \frac{2880}{72} \]
5. Perform the division: \[ x = 40 \]

So, the value of x is 40.

Verifying the Solution

We can quickly check if the harmonic mean of 60 and 40 is indeed 48:

\[ \text{HM}(60, 40) = \frac{2 \times 60 \times 40}{60 + 40} = \frac{2 \times 2400}{100} = \frac{4800}{100} = 48 \]

The calculation confirms that our value of x = 40 is correct.

Revision Table: Types of Means

Additional Information: Relationship Between Means

For any set of positive numbers, the following relationship always holds true:

\[ \text{AM} \ge \text{GM} \ge \text{HM} \]

Equality holds only when all the numbers in the set are equal.

For two positive numbers 'a' and 'b', there's also a relationship between AM, GM, and HM:

\[ \text{GM}^2 = \text{AM} \times \text{HM} \]

This means the geometric mean is the geometric mean of the arithmetic and harmonic means.

Understanding these different types of means and their relationships is crucial for various mathematical and statistical applications.