Finding the Ratio of Two Numbers Given Their Sum and Difference

The process of determining the ratio of two numbers x and y given their sum a and difference b can be approached algebraically. This method involves solving a system of linear equations to find the individual values of x and y, and subsequently calculating the ratio x/y.

Algebraic Approach

Consider two numbers x and y. Given that:

These two equations can be manipulated to find the individual values of x and y.

Step-by-Step Solution

Start by adding the two equations:

x y x - y a b

2x a b

x (a b) / 2

Next, subtract the second equation from the first:

x y - x y a - b

2y a - b

y (a - b) / 2

With the values of x and y obtained, the ratio x/y can be computed:

x/y [(a b) / 2] / [(a - b) / 2] (a b) / (a - b) if a ≠ b

In the special case where a b:

Substituting a for b in the sum equation:

x y a

x - y a

Add these equations to get:

2x 2a

x a

Subtract the second from the first to get:

2y 0

y 0

The ratio x/y is undefined (infinity), as y 0:

x/y a/0 ∞

Example Calculation

Let us illustrate the process with an example where a sum and difference are given:

A B 8 (a)

A - B 5 (b)

Add the two equations:

2A 13

A 6.5

Subtract the second from the first:

2B 3

B 1.5

The ratio A/B is:

A/B 6.5 / 1.5 13 / 3

Summary

By solving a system of linear equations, we can effectively determine the individual values of x and y from their sum and difference, and subsequently calculate the desired ratio. The process is straightforward and relies on basic algebraic operations.

Keyword Optimization

This content is optimized with the keywords: ratio of two numbers, given sum and difference, and algebraic equations to ensure it ranks well in search engine results.