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Cross multiplication is a powerful technique used to solve proportions easily. When dealing with a fraction equal to another fraction, you can find the value of the unknown by multiplying the numerator of one fraction by the denominator of the other fraction and setting them equal. This method simplifies the process and removes the fractions.
In the exercise, we had the proportion \(\frac{x}{5} = \frac{12}{3}\).
Cross multiplication involves multiplying the numerator of the left fraction by the denominator of the right fraction, and the numerator of the right fraction by the denominator of the left fraction.
The step looks like this:

• \text{x \times 3 = 5 \times 12}\text

This yields \(3x = 60\). Now, solving for \x\ becomes straightforward.

Simplifying fractions is a crucial step when solving proportions, as it makes calculations easier and reduces errors. Simplifying involves finding an equivalent fraction where the numerator and denominator have no common factors other than 1.
Look at the given fraction on the right-hand side of the proportion: \(\frac{12}{3}\).
By dividing both the numerator and the denominator by their greatest common divisor, which is 3 in this case, we get:

• $$ \frac{12}{3} = 4$$

This reveals a simpler form, making it easier to work with. Now the proportion is \(\frac{x}{5} = 4\), ready for the next steps like cross multiplication.

Once we have simplified the fractions and cross-multiplied, we are often left with a linear equation. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations can be solved using straightforward algebraic manipulation.
From the cross multiplication step, we had \(3x = 60\).
Solving for \x\ involves isolating the variable on one side of the equation. Divide both sides by the coefficient of \x\ which is 3:

• \text{3x = 60}\text
• \text{\x = \frac{60}{3} = 20}\text

Now we have the solution: \(x = 20\).
This highlights that understanding linear equations and how to manipulate them is vital in solving proportions effectively.