Question

\(\frac{36}{3}=\frac{12}{x}\)

Short answer

The value of \( x \) is 1.

Step-by-step solution

Understand the Equation

The given equation is a proportion: \( \frac{36}{3} = \frac{12}{x} \). This means the ratios of two fractions are equal, and our goal is to find the value of \( x \).

Simplify the Left Side

Begin by simplifying the left side of the equation: \( \frac{36}{3} = 12 \), since 36 divided by 3 equals 12.

Set the Simplified Equation

Rewrite the equation with the simplified left side: \( 12 = \frac{12}{x} \). This now tells us that 12 is equal to the fraction on the right.

Solve for x

To find \( x \), set up the equation: \( 12x = 12 \). By doing cross-multiplication (multiplying both sides of the equation by \( x \)), we isolate \( x \) on one side.

Isolate x

Divide both sides of the equation by 12 to solve for \( x \): \( x = 1 \).

Verification

Verify the solution by substituting \( x = 1 \) back into the original proportion equation: \( \frac{36}{3} = \frac{12}{1} \). Simplifying both, we get 12 on each side, confirming our solution.

Key concepts

Cross Multiplication

A powerful tool when dealing with proportions, cross multiplication helps simplify and solve equations involving two fractions. When two fractions are set equal to each other, as in a proportion, you can cross multiply to clear the fractions. Specifically, you multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction.
For example, with the equation \( \frac{36}{3} = \frac{12}{x} \), you cross multiply to obtain the equation \( 36x = 36 \). This step allows you to transition from a complex fraction equation to a simpler equation, ready for solving.

• Cross multiplication works because it maintains the equality by multiplying each side of the equation by the denominators, effectively "clearing" them.
• This method is particularly effective with proportions because it directly isolates one variable in terms of known values.

Solving Equations

The art of solving equations is all about finding the unknown variable. It requires isolating the variable on one side of the equation. Once you've used cross multiplication, you will often get a much simpler equation. In our example, after cross multiplying, it simplifies to \( 36x = 36 \).
To isolate \( x \), you perform inverse operations to manipulate the equation.

• Start by identifying the operation applied to \( x \). Here it's a multiplication, so you'll want to use division to undo it.
• You divide both sides of the equation by the same number to maintain equality. This gives \( x = 1 \), as \( \frac{36}{36} = 1 \).

This solution shows that once proportions are simplified into simple equations, finding the unknown becomes straightforward.

Simplifying Fractions

Simplifying fractions is a crucial skill, making it easier to work with numbers. It involves reducing fractions to their simplest form where the numerator and the denominator have no common factors other than 1. For our problem, simplifying \( \frac{36}{3} \) to \( 12 \) is an important first step.
Here's how you do it:

• Divide the numerator and the denominator by their greatest common factor (GCF). For 36 and 3, the GCF is 3.
• This simplification results in smaller, more manageable numbers, transforming \( \frac{36}{3} \) into \( 12 \).

This simplification not only simplifies the equation but also aids in verifying the final solution. Simplified fractions make calculations quicker and less prone to errors, ensuring you easily verify that both sides of the equation are equal upon solving.