Question

Challenge Problem Harmonic Mean For \(0

Short answer

The inequalities \( a < h \) and \( h < b \) hold, so \( a < h < b \).

Step-by-step solution

- Write the harmonic mean formula

Given the formula for the harmonic mean: \[ \frac{1}{h} = \frac{1}{2} \left( \frac{1}{a} + \frac{1}{b} \right) \]

- Simplify the right-hand side

Simplify the right-hand side of the equation: \[ \frac{1}{h} = \frac{1}{2} \left( \frac{a + b}{ab} \right) = \frac{a + b}{2ab} \]

- Solve for h

By taking the reciprocal of both sides, we get: \[ h = \frac{2ab}{a+b} \]

- Show that a < h

To show that \( a < h \), consider: \[ a < \frac{2ab}{a+b} \] Multiply both sides by \( (a + b) \): \[ a(a + b) < 2ab \] This simplifies to: \[ a^2 + ab < 2ab \] Subtract \( ab \) from both sides: \[ a^2 < ab \] Divide by \( a \) (which is positive): \[ a < b \] This inequality holds since \( a < b \).

- Show that h < b

To show that \( h < b \), consider: \[ \frac{2ab}{a+b} < b \] Multiply both sides by \( (a + b) \): \[ 2ab < b(a + b) \] This simplifies to: \[ 2ab < ab + b^2 \] Subtract \( ab \) from both sides: \[ ab < b^2 \] Divide by \( b \) (which is positive): \[ a < b \] This inequality holds since \( a < b \).

Key concepts

Inequalities

In mathematics, an inequality is a relation that holds between two values when they are different. Inequalities are foundational in diverse areas such as

• Algebra
• Calculus
• Optimization Problems

In this context, we were tasked with showing that one number (the harmonic mean) lies strictly between two other numbers (\(0 < a < b \)).
To achieve this, we use manipulation to establish the inequalities \( a < h \) and \( h < b \). These steps help us validate the placement of the harmonic mean between given values, ensuring it accurately captures the intended mean.

Algebraic Manipulation

Algebraic manipulation is a process of rearranging equations or expressions to solve for a variable or simplify an expression. For the harmonic mean problem, we encountered several stages of algebraic manipulation.
Initially, we simplified the given harmonic mean formula: \[ \frac{1}{h} = \frac{1}{2} \left( \frac{1}{a} + \frac{1}{b} \right) \].
Breaking this down, we get: \( \frac{1}{h} = \frac{a + b}{2ab} \), and by finding the reciprocal, \( h = \frac{2ab}{a + b} \). This crucial step transforms the problem into a form where inequalities can be validated.
Further manipulation displayed how \( a < \frac{2ab}{a + b} \) and \( \frac{2ab}{a + b} < b \), substantiating that \( a < h < b \). Understanding each algebraic step profoundly helps to elucidate the role and placement of each variable in the solution.

Reciprocals

A reciprocal of a number is one divided by that number, often used to simplify fractions or solve equations. For the harmonic mean problem, reciprocals are central to transforming and simplifying the harmonic mean formula.

• The original equation: \( \frac{1}{h} = \frac{1}{2} \left( \frac{1}{a} + \frac{1}{b} \right) \) involves reciprocals of \( a \) and \( b \). Taking reciprocals helps compress and solve expressions.
• After simplifying, we get \( h = \frac{2ab}{a + b} \), where reciprocation played a pivotal role.

In summary, understanding reciprocals is vital for manipulating complex algebraic expressions. They offer a convenient method to swap numerators and denominators, essential for simplification and solving problems like the one discussed.