(a+b)/(c+d) Inequality

5 min read Jun 16, 2024

Understanding the (a+b)/(c+d) Inequality

The inequality (a+b)/(c+d) is a fundamental concept in mathematics, particularly in the realm of inequalities. It states that for positive real numbers a, b, c, and d, where a > b and c > d, the following holds:

(a + b)/(c + d) < a/c

This inequality has various applications and helps establish relationships between fractions and their components.

Intuitive Explanation

Imagine two fractions, a/c and b/d, where a > b and c > d. The inequality (a+b)/(c+d) < a/c essentially states that the average of these two fractions is always smaller than the larger fraction.

To visualize this, consider the following:

a/c represents a larger "piece" of a whole.
b/d represents a smaller "piece" of a whole.
(a+b)/(c+d) represents the average size of these two "pieces" after combining them.

Since we are combining a larger piece with a smaller piece, the resulting average size will always be closer to the smaller piece (b/d) and hence smaller than the larger piece (a/c).

Proof of the Inequality

We can formally prove the inequality using basic algebraic manipulations:

Start with the given conditions: a > b and c > d.
Subtract b from both sides of the first inequality: a - b > 0.
Subtract d from both sides of the second inequality: c - d > 0.
Multiply the inequalities from steps 2 and 3: (a - b)(c - d) > 0.
Expand the product: ac - ad - bc + bd > 0.
Add ad and bc to both sides: ac + bd > ad + bc.
Divide both sides by (c + d)(c): (ac + bd)/(c + d)(c) > (ad + bc)/(c + d)(c).
Simplify: (a/c) + (bd/c(c+d)) > (ad/c(c+d)) + (b/d).
Subtract (bd/c(c+d)) from both sides: (a/c) > (ad/c(c+d)) + (b/d) - (bd/c(c+d)).
Simplify the right side: (a/c) > (a/c + b/d)(d/c + d/c).
Since (d/c + d/c) > 1, we can conclude: (a/c) > (a/c + b/d).
Therefore: (a+b)/(c+d) < a/c.

Applications

The (a+b)/(c+d) inequality has various applications in different areas of mathematics, including:

Calculus: It is used in proving certain inequalities involving derivatives and integrals.
Number Theory: It can be applied to establish relationships between fractions and their components in number theory problems.
Probability: The inequality plays a role in proving certain results in probability theory, particularly those involving conditional probabilities.

Conclusion

The (a+b)/(c+d) inequality is a powerful tool for comparing fractions and establishing relationships between their components. Its intuitive explanation and rigorous proof make it a valuable concept in various mathematical disciplines. Understanding this inequality can lead to deeper insights and facilitate the solution of complex mathematical problems.