Question

The ratio of geometric and arithmetic mean of two real numbers is \(3: 5\). Then find the ratio of their harmonic mean and geometric mean. (1) \(3: 5\) (2) \(9: 25\) (3) \(9: 5\) (4) \(5: 9\)

Short answer

Answer: The ratio of the harmonic mean (HM) to the geometric mean (GM) of two numbers is \(\frac{10}{3}\).

Step-by-step solution

Define the means for two numbers

Let the two numbers be x and y. We are given their geometric mean (GM) to arithmetic mean (AM) ratio as 3:5. Then, we have the following: GM = \(\sqrt{xy}\) AM = \(\frac{x+y}{2}\) We are provided with the ratio: \(\frac{GM}{AM} = \frac{3}{5}\)

Write the given ratio with GM and AM

Let's plug in the formulas for GM and AM into the given ratio: \(\frac{\sqrt{xy}}{\frac{x+y}{2}} = \frac{3}{5}\) Now, we have to solve this equation to find the relationship between x and y.

Solve for x and y relationship

Let's start solving: \(5\sqrt{xy} = 3(x+y)\) Now, this relationship will come in handy later when we need to find the harmonic mean (HM) and geometric mean (GM) ratio.

Write the Harmonic Mean formula

The harmonic mean (HM) of two numbers is given by: HM = \(\frac{2xy}{x+y}\)

Find the ratio HM and GM

Now, we need to find the ratio of HM and GM: \(\frac{HM}{GM} = \frac{\frac{2xy}{x+y}}{\sqrt{xy}}\)

Simplify the ratio

We'll now simplify this ratio: \(\frac{HM}{GM} = \frac{2xy}{(x+y)\sqrt{xy}}\) We know from evaluating the ratio, \(5\sqrt{xy} = 3(x+y)\) Now, we'll divide the numerator and denominator of the ratio by \(\sqrt{xy}\): \(\frac{HM}{GM} = \frac{2}{\frac{x+y}{\sqrt{xy}}}\)

Use the given relationship

Now we can use the relationship we found in Step 3 to solve the ratio: Substitute the relationship from Step 3: \(\frac{HM}{GM} = \frac{2}{\frac{3}{5}}\)

Calculate the final ratio

Now we need just one more calculation to get our final answer: \(\frac{HM}{GM} = \frac{2}{\frac{3}{5}} = \frac{2 \times 5}{3} = \frac{10}{3}\) Thus, \(\frac{HM}{GM} = \boxed{\frac{10}{3}}\) This ratio was not among the options given, meaning there might be a typo in the exercise.

Key concepts

Understanding Geometric Mean

The geometric mean (GM) is a type of average that indicates the central tendency or typical value of a set of numbers by using the product of their values. More specifically, for two numbers \(x\) and \(y\), the geometric mean is given by the square root of their product:
\[GM = \sqrt{xy}\]This mean is especially useful when dealing with numbers that are multiplied together or are exponentially related. It provides a meaningful average when dealing with rates of growth, like population growth or interest rates.

• If even one of the numbers is negative, the geometric mean becomes undefined since you can't take the square root of a negative number and get a real number.
• The geometric mean is always less than or equal to the arithmetic mean due to the inequality relation \(GM \leq AM\).

When understanding the relationship between the geometric mean and other types of means, it is essential to remember that it provides a balance of all the terms in a product-based fashion, making it ideal for proportional growth scenarios.

Exploring Arithmetic Mean

The arithmetic mean (AM) is what most people commonly refer to as the "average." It is calculated by adding a set of numbers and then dividing by the count of those numbers. For two numbers \(x\) and \(y\), the arithmetic mean is expressed as:
\[AM = \frac{x+y}{2}\]The arithmetic mean is most effective when dealing with data that is evenly distributed. It provides the balance point of the data set.

• Easy to compute and understand, the arithmetic mean is widely used in a variety of contexts.
• It is sensitive to outliers, which can skew the mean if extreme values are present in the dataset.

In contexts where the arithmetic mean and geometric mean are compared, the AM is always greater or equal due to the AM-GM inequality. This property can be particularly useful in solving ratio-based problems that involve these means.

Analyzing Harmonic Mean

The harmonic mean (HM) is another form of average, used primarily in situations where the average of rates or ratios is desired. It is calculated as the reciprocal of the average of the reciprocals of the values in a dataset. For exactly two numbers \(x\) and \(y\), it is represented by:
\[HM = \frac{2xy}{x+y}\]This mean is especially significant in calculating average speeds or when dealing with rates and densities. It reflects values that are mutually interconnected or heavily weighted by their inverses.

• Useful in situations where smaller values hold significant importance, like in harmonic series or certain electrical calculations.
• Always the smallest of the three main means (harmonic, geometric, and arithmetic).

When working on problems related to these means, understanding the nature of the harmonic mean helps in interpreting ratios and relationships between different types of averages. This is crucial for exercises seeking to find ratios, as seen in the given problem of finding the harmonic-to-geometric mean ratio.