Problem 13 - Elementary and Intermediate Algebra

Question

$$ \text { Solve for } x: \frac{12}{18}=\frac{x}{24} $$

Step-by-Step Solution

Answer

x = 16.

Step 1: Set the Proportions Equal

We need to solve the equation \(\frac{12}{18} = \frac{x}{24}\) for \(x\). These are two fractions that are equal to each other, known as a proportion.

Step 2: Cross-Multiply

Cross-multiply to eliminate the fractions, which means multiplying diagonally across the equal sign: \(12 \times 24 = 18 \times x\).

Step 3: Simplify the Multiplications

Calculate \(12 \times 24\). This equals 288. So, the equation becomes \(288 = 18x\).

Step 4: Solve for x

Divide both sides of the equation by 18 to solve for \(x\): \(x = \frac{288}{18}\).

Step 5: Simplify the Result

Calculate \(\frac{288}{18}\). You can simplify this by dividing both the numerator and the denominator by 6. \(\frac{288 \div 6}{18 \div 6} = \frac{48}{3} = 16\), so \(x = 16\).

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a method used to solve equations involving proportions. A proportion is simply two fractions set equal to each other. The basic idea is to eliminate the fractions by multiplying the numerator (top part of the fraction) of each fraction by the denominator (bottom part of the fraction) of the other fraction across the equal sign.
For example, in the equation \( \frac{12}{18} = \frac{x}{24} \), cross-multiplication involves:

• Multiplying 12 (numerator of the first fraction) by 24 (denominator of the second fraction) to get one product.
• Multiplying 18 (denominator of the first fraction) by \(x\) (numerator of the second fraction) to get another product.

This results in the equation \(12 \times 24 = 18 \times x\). Now, the fractions are removed, leaving a simple equation. This powerful tool makes it easier to work with proportions without the messiness of fractions.

The Process of Solving Equations

Solving equations is a foundational skill in algebra. It involves finding the value of the variable that makes the equation true. Once you've used cross-multiplication to convert a proportion into a simple equation, solving for the variable is straightforward.
Take the equation from our earlier cross-multiplication example: \(288 = 18x\). To solve for \(x\), you need to isolate it on one side of the equation. Here, division is your ally:

• First, divide both sides of the equation by 18, which is the coefficient (number in front) of \(x\).
• This gives you \(x = \frac{288}{18}\).

By dividing, you're effectively simplifying the equation, making it manageable. Solving equations can be fun and rewarding once you become comfortable with the necessary steps and operations.

Demystifying Fractions

Fractions are a fundamental part of mathematics, representing parts of a whole or a ratio between two numbers. Understanding fractions is crucial when dealing with proportions and cross-multiplication.
In the context of solving proportions like \(\frac{12}{18} = \frac{x}{24}\):

• The numerator (the top number) represents how many parts we have.
• The denominator (the bottom number) indicates the total number of parts into which the whole is divided.

When you simplify fractions, you make them easier to work with by expressing them in their simplest form. For instance, simplifying \(\frac{288}{18}\) involves finding the greatest common divisor (GCD) of 288 and 18, which is 6, and dividing both numbers by this value.
Once you've simplified the fraction, you get \(\frac{48}{3} = 16\), confirming that \(x = 16\). Understanding and manipulating fractions are key to solving many mathematical problems efficiently.