Finding the x^2/y^2 Ratio from a Given Equation

The question asks us to find the value of the ratio \(x^2 / y^2\) given the equation \(\frac{6x + 4y}{6x - 4y} = \frac{8}{6}\). To solve this, we need to find the ratio of \(x\) to \(y\) first, and then square it.

Step-by-Step Calculation of the x/y Ratio

Let's start with the given equation:

\(\frac{6x + 4y}{6x - 4y} = \frac{8}{6}\)

First, we can simplify the fraction on the right side:

\(\frac{8}{6} = \frac{4}{3}\)

So the equation becomes:

\(\frac{6x + 4y}{6x - 4y} = \frac{4}{3}\)

Now, we can use cross-multiplication to eliminate the denominators. Multiply the numerator of the left side by the denominator of the right side, and set it equal to the product of the denominator of the left side and the numerator of the right side:

\(3 \times (6x + 4y) = 4 \times (6x - 4y)\)

Next, distribute the numbers on both sides of the equation:

\(18x + 12y = 24x - 16y\)

Now, we need to group the terms with \(x\) on one side and the terms with \(y\) on the other side. Let's move the \(12y\) term to the right side by subtracting \(12y\) from both sides, and move the \(24x\) term to the left side by subtracting \(24x\) from both sides (or equivalently, move \(18x\) to the right and \(-16y\) to the left):

\(12y + 16y = 24x - 18x\)

Combine the like terms on both sides:

\(28y = 6x\)

Now we want to find the ratio \(x/y\). To do this, we can divide both sides by \(y\) and then by 6:

\(\frac{28y}{y} = \frac{6x}{y}\)

\(28 = \frac{6x}{y}\)

Now, divide both sides by 6:

\(\frac{28}{6} = \frac{x}{y}\)

Simplify the fraction \(\frac{28}{6}\) by dividing both the numerator and denominator by their greatest common divisor, which is 2:

\(\frac{28 \div 2}{6 \div 2} = \frac{14}{3}\)

So, the ratio \(x/y\) is:

\(\frac{x}{y} = \frac{14}{3}\)

Calculating the Value of x^2/y^2

The question asks for the value of \(x^2 / y^2\). We can find this by squaring the ratio \(x/y\) that we just found:

\(\frac{x^2}{y^2} = \left(\frac{x}{y}\right)^2\)

Substitute the value of \(\frac{x}{y}\) we found:

\(\frac{x^2}{y^2} = \left(\frac{14}{3}\right)^2\)

Now, square the numerator and the denominator:

\(\left(\frac{14}{3}\right)^2 = \frac{14^2}{3^2} = \frac{196}{9}\)

So, the value of \(x^2 / y^2\) is \(\frac{196}{9}\).

Revision Table: Key Concepts for Ratio and Proportion Problems

Additional Information: Alternative Methods (Componendo and Dividendo)

This problem can also be solved using the Componendo and Dividendo rule, which states that if \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a+b}{a-b} = \frac{c+d}{c-d}\). Let \(a = 6x + 4y\) and \(b = 6x - 4y\). Also, let \(c = 8\) and \(d = 6\).

Using the rule:

\(\frac{(6x + 4y) + (6x - 4y)}{(6x + 4y) - (6x - 4y)} = \frac{8 + 6}{8 - 6}\)

Simplify the numerator and denominator on both sides:

\(\frac{6x + 4y + 6x - 4y}{6x + 4y - 6x + 4y} = \frac{14}{2}\)

\(\frac{12x}{8y} = 7\)

Now, find the ratio \(x/y\):

\(\frac{x}{y} = 7 \times \frac{8}{12}\)

\(\frac{x}{y} = 7 \times \frac{2}{3}\)

\(\frac{x}{y} = \frac{14}{3}\)

This gives the same ratio as before. Squaring this ratio yields the same result for \(x^2/y^2\):

\(\frac{x^2}{y^2} = \left(\frac{14}{3}\right)^2 = \frac{196}{9}\)

This shows that understanding different algebraic techniques, like Componendo and Dividendo, can provide alternative paths to the solution in ratio and proportion problems.