Question

\(\frac{30-x}{x}=\frac{3}{2}\)

Short answer

x = 12.

Step-by-step solution

Understand the Equation

We are tasked with solving the equation \( \frac{30-x}{x} = \frac{3}{2} \). This is a rational equation where the variable \( x \) is involved in both the numerator and the denominator of the left side.

Eliminate the Fraction

To eliminate the fraction, we will cross-multiply. This gives us: \[ (30-x) \times 2 = 3 \times x \] This step simplifies the equation to one without fractions.

Simplify the Equation

Expand and simplify the equation from Step 2: \[ 60 - 2x = 3x \] We will bring all terms involving \( x \) to one side and constant terms to the other.

Isolate the Variable

Add \(2x\) to both sides to collect all terms involving \( x \) on one side: \[ 60 = 3x + 2x \] Simplify the right side: \[ 60 = 5x \]

Solve for x

To isolate \( x \), divide both sides of the equation by 5: \[ x = \frac{60}{5} \] Simplify to find \( x = 12 \).

Key concepts

Cross-Multiplication

Cross-multiplication is an algebraic method used to eliminate fractions in equations by multiplying the denominators across the equation. This approach makes solving equations much simpler, especially when dealing with rational expressions.
To apply cross-multiplication, you take the numerator of one fraction and multiply it by the denominator of the other fraction. This product equals the product of the other numerator and its respective denominator. In the equation \( \frac{30-x}{x} = \frac{3}{2} \), cross-multiplying involves:

• Multiply \((30-x)\) by 2.
• Multiply \(3\) by \(x\).

This gives us the equation: \[ (30-x) \times 2 = 3 \times x \]By using cross-multiplication, we effectively eliminate all fractions and work with a simple linear equation. This makes further manipulations, such as isolating variables, much easier.

Isolating Variables

Isolating variables is a crucial technique for solving equations, where we aim to have the unknown variable by itself on one side of the equation. This process often requires the rearrangement of terms and the use of inverse operations.
Once we have cleared the fractions using cross-multiplication, our equation is:\[ 60 - 2x = 3x \] To isolate the variable \(x\), we perform the following steps:

• Move all terms involving \(x\) to one side of the equation. This often involves adding or subtracting terms from both sides. In this case, add \(2x\) to both sides:
\[ 60 = 3x + 2x \]

After this step, we combine the \(x\) terms to simplify:\[ 60 = 5x \]Now, the variable \(x\) is isolated, and we can solve for it by dividing both sides by \(5\). This ensures that \(x\) sits perfectly on one side of the equation, making it clear and easy to solve.

Simplifying Equations

Simplifying equations helps us reduce the complexity of mathematical expressions and make them easier to solve. In our example, this process involves several steps, from expanding terms to combining like terms.
After cross-multiplying, we have:\[ (30-x) \times 2 = 3x \] Expanding the left side yields:\[ 60 - 2x = 3x \]At this point, it's crucial to group all terms involving the variable on one side. We do this by repositioning terms and combining like terms:

• Move \(2x\) to the other side to add to \(3x\).
• Simplify the equation to \(60 = 5x\).

Finally, dividing both sides by \(5\) gives us the simplified solution:\[ x = 12 \]Simplifying not only makes the equation easier to understand, but it also reveals the clear path to finding the solution.