Problem 65 - A Complete Resource Book in Mathematics for JEE Main

Question

Let the harmonic mean and the geometric mean of two positive numbers be in the ratio \(4: 5\). The two numbers are in the ratio (A) \(1: 1\) (B) \(2: 1\) (C) \(3: 1\) (D) \(4: 1\)

Step-by-Step Solution

Answer

The two numbers are in the ratio \(4:1\).

Step 1: Understand the definitions

The harmonic mean (HM) of two numbers, say \(a\) and \(b\), is given by \(HM = \frac{2ab}{a+b}\). \ The geometric mean (GM) is given by \(GM = \sqrt{ab}\). This forms the basis of our solution.

Step 2: Set up the equation using given ratios

According to the problem, the ratio of the harmonic mean to the geometric mean is 4:5. \[\frac{HM}{GM} = \frac{4}{5}.\] Substitute the expressions for HM and GM:\[\frac{\frac{2ab}{a+b}}{\sqrt{ab}} = \frac{4}{5}.\]

Step 3: Simplify the equation

Simplify the left side of the equation:\[\frac{2ab}{(a+b)\sqrt{ab}} = \frac{4}{5}.\]Simplifying gives:\[\frac{2\sqrt{ab}}{a+b} = \frac{4}{5}.\]

Step 4: Clear the fraction by cross-multiplying

Cross-multiply to eliminate the fractions:\[5 \times 2\sqrt{ab} = 4(a+b).\]Thus,\[10\sqrt{ab} = 4a + 4b.\]

Step 5: Simplify and express in terms of one variable

Factor out the common term on the right:\[10\sqrt{ab} = 4(a+b).\]Set\(a = kb\) where \(k\) is the ratio that we want to find.Substitute \(a = kb\) into the equation:\[10\sqrt{k}\, b = 4(kb + b). \]

Step 6: Solve for the ratio k

Simplify:\[10\sqrt{k}b = 4(k + 1)b.\]Divide both sides by \(b\):\[10\sqrt{k} = 4(k + 1).\]Solve for \(k\) to find the ratio of \(a\) and \(b\):\[5\sqrt{k} = 2(k + 1). \]Squaring both sides gives:\[25k = 4(k + 1)^2.\] Expanding and rearranging results in:\[4k^2 - 17k + 4 = 0.\]

Step 7: Solve the quadratic equation for k

Solve \(4k^2 - 17k + 4 = 0\) using the quadratic formula, \(k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -17\), \(c = 4\):\[k = \frac{17 \pm \sqrt{(-17)^2 - 4\cdot4\cdot4}}{2\times4} = \frac{17 \pm \sqrt{289 - 64}}{8} = \frac{17 \pm \sqrt{225}}{8} = \frac{17 \pm 15}{8}.\] Therefore, \(k = 4\) or \(k = \frac{1}{8}\). Since \(k = \frac{a}{b}\), and we seek whole number options, \(k = 4:1\).

Step 8: Conclusion

Thus \(a : b = 4 : 1\). Hence, the two numbers are in the ratio \(4:1\).

Key Concepts

Harmonic Mean

When discussing means, the harmonic mean is particularly useful when dealing with rates or ratios. For two positive numbers, the harmonic mean (HM) is expressed using the formula:

• \( HM = \frac{2ab}{a+b} \)

This formula reflects the idea that the harmonic mean tends to be dominated by smaller values of the dataset. For instance, in problems involving speeds, where different legs of a trip are traveled at different speeds, the harmonic mean provides a more accurate average speed than the arithmetic mean.
Understanding the harmonic mean becomes crucial in exercises involving proportions, as it relates inversely to the principle of the arithmetic mean, tending toward the lower end of the values. In the context of our exercise, the relationship between the harmonic mean and the geometric mean helps establish a specific ratio necessary to solve for unknown numbers.

Geometric Mean

The geometric mean is another important type of mean, used especially in cases of exponential growth or proportional change. For two positive numbers, the geometric mean (GM) is calculated as:

• \( GM = \sqrt{ab} \)

Geometric means are particularly significant in scenarios involving multiplicative processes, such as compound interest or the scaling of variables in geometrical applications. This mean provides an efficient way to find a central tendency where the numbers are continuously interdependent or fluctuate by percentages.
In the provided exercise, the geometric mean, coupled with the harmonic mean, offers an insightful way to infer relationships between the two numbers. By exploring their ratio, we can delve deeper into mathematical properties that guide us to find specific proportions between numerical values.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation that typically appears in the form:

• \( ax^2 + bx + c = 0 \)

To solve such an equation, you can use methods like factoring, completing the square, or the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula is reliable and allows us to find the roots of any quadratic equation quickly.
In our scenario, we used the quadratic formula to determine a variable \( k \) that reflects the ratio of the two numbers. Solving the quadratic equation derived from the relationship between the harmonic and geometric means helped isolate potential values of \( k \), guiding us to the correct solution in the context of the ratios involved.

Ratio of Numbers

Ratios are expressions that compare quantities relative to each other and are foundational in many aspects of mathematical analysis. In a problem-solving context, understanding the ratio of two numbers helps reveal their relative magnitudes or contributions one to the other.

• If the ratio of two numbers \( a \) and \( b \) is represented as \( a:b \), it can also be expressed as \( \frac{a}{b} \).

Analyzing ratios provides insight into how one variable scales in relation to another. It's crucial, especially in problems where relationships are quantified through proportions or averages, such as means.
Ultimately, within our exercise, determining the ratio of numbers involved leveraging both the harmonic and geometric means. By establishing a relationship between these two means, we found the actual proportion governing the two unknown numbers in question, leading us to decipher their specific ratio efficiently.