Solving Ratio Problems: From (x+y):(x-y) to (x²+y²):(x²-y²)

This problem involves working with ratios and algebraic expressions. We are given a ratio relating \(x\) and \(y\) and asked to find a different ratio involving squares of \(x\) and \(y\).

Understanding the Given Ratio

The initial ratio is given as \( (x + y) : (x - y) = 3 : 2 \). This can be written as a fraction:

\( \frac{x + y}{x - y} = \frac{3}{2} \)

Finding the Relationship Between x and y

To find the relationship between \(x\) and \(y\), we can cross-multiply the fractional equation:

\( 2(x + y) = 3(x - y) \)

Now, distribute the numbers on both sides:

\( 2x + 2y = 3x - 3y \)

Gather the terms with \(x\) on one side and terms with \(y\) on the other side:

\( 2y + 3y = 3x - 2x \)

\( 5y = x \)

This equation \(x = 5y\) tells us the relationship between \(x\) and \(y\). It means that \(x\) is five times the value of \(y\). We can also write this as a ratio \(x : y = 5 : 1\).

Calculating the Required Ratio

We need to find the ratio \( (x^{2} + y^{2}) : (x^{2} - y^{2}) \). We can substitute \(x = 5y\) into this expression.

\( \frac{x^{2} + y^{2}}{x^{2} - y^{2}} \)

Substitute \(x = 5y\) into the numerator and the denominator:

\( \frac{(5y)^{2} + y^{2}}{(5y)^{2} - y^{2}} \)

Calculate the squares:

\( \frac{25y^{2} + y^{2}}{25y^{2} - y^{2}} \)

Combine like terms in the numerator and the denominator:

\( \frac{(25 + 1)y^{2}}{(25 - 1)y^{2}} = \frac{26y^{2}}{24y^{2}} \)

Assuming \(y \neq 0\) (because if \(y=0\), then \(x=5(0)=0\), making the original ratio \(0:0\) which is undefined), we can cancel out \(y^{2}\) from the numerator and the denominator:

\( \frac{26}{24} \)

Simplifying the Final Ratio

The fraction \( \frac{26}{24} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

\( \frac{26 \div 2}{24 \div 2} = \frac{13}{12} \)

So, the ratio \( (x^{2} + y^{2}) : (x^{2} - y^{2}) \) is \( 13 : 12 \).

Summary of Steps

1. Convert the given ratio into an equation.
2. Solve the equation to find the relationship between \(x\) and \(y\).
3. Substitute the relationship into the expression for the required ratio.
4. Simplify the expression to get the final ratio.

The ratio \( (x^{2} + y^{2}) : (x^{2} - y^{2}) \) is \( 13 : 12 \).

Revision Table: Key Concepts

Additional Information: Componendo and Dividendo

This problem can also be solved using the Componendo and Dividendo rule, which is derived from proportions.

If \( \frac{a}{b} = \frac{c}{d} \), then \( \frac{a+b}{a-b} = \frac{c+d}{c-d} \).

Given \( \frac{x + y}{x - y} = \frac{3}{2} \). Here, \(a = x+y\) and \(b = x-y\), and \(c=3\) and \(d=2\). However, the structure doesn't directly match \(a/b = c/d\). Let's rearrange the initial ratio slightly by considering \(a=x\) and \(b=y\). The given ratio is \( \frac{x+y}{x-y} = \frac{3}{2} \).

We can apply Componendo and Dividendo to the expression \( \frac{x}{y} \). Let's first find \(x/y\) from the given equation \(x=5y\), which gives \( \frac{x}{y} = \frac{5}{1} \).

Now, consider the expression we need to find the ratio for: \( \frac{x^{2} + y^{2}}{x^{2} - y^{2}} \). Divide both numerator and denominator by \(y^{2}\) (assuming \(y \neq 0\)):

\( \frac{\frac{x^{2}}{y^{2}} + \frac{y^{2}}{y^{2}}}{\frac{x^{2}}{y^{2}} - \frac{y^{2}}{y^{2}}} = \frac{(\frac{x}{y})^{2} + 1}{(\frac{x}{y})^{2} - 1} \)

Substitute \( \frac{x}{y} = \frac{5}{1} \) into this expression:

\( \frac{(\frac{5}{1})^{2} + 1}{(\frac{5}{1})^{2} - 1} = \frac{5^{2} + 1}{5^{2} - 1} \)

\( \frac{25 + 1}{25 - 1} = \frac{26}{24} = \frac{13}{12} \)

This method also gives the same result and demonstrates another way to approach ratio problems involving squares of variables after finding the basic ratio of the variables.