Question

Solve each proportion. \(\frac{x}{3}=\frac{12}{18}\)

Short answer

The value of \( x \) is 2.

Step-by-step solution

Cross-Multiply

Proportions can be solved using the cross-multiplication method. Here, multiply both sides of the equation's numerator by the opposite side's denominator. Thus, multiply 18 by \( x \) and 3 by 12. This gives the equation:\[ 18x = 3 imes 12 \]

Simplify Right Side

Calculate the product on the right-hand side of the equation: \( 3 imes 12 = 36 \). This updates our equation to:\[ 18x = 36 \]

Solve for x

To isolate \( x \), divide both sides of the equation by 18, which is the coefficient of \( x \):\[ x = \frac{36}{18} \]

Simplify the Fraction

Simplify the fraction \( \frac{36}{18} \). Since 36 divided by 18 equals 2, we find:\[ x = 2 \]

Key concepts

Cross-Multiplication

Cross-multiplication is an essential technique for solving equations involving proportions. A proportion is simply two ratios that are considered equal, much like in the exercise with \( \frac{x}{3} = \frac{12}{18} \). This technique shines because it helps eliminate fractions, which are often perceived as complex.When you cross-multiply, you simply multiply each numerator by the opposite denominator. This concept is akin to creating a balance by crossing across the equal sign. For instance:

• Multiply 18 and \( x \) from the first ratio, and 3 and 12 from the second, resulting in the equation: \( 18x = 3 \times 12 \).
• This step converts a single equation with fractions into an equation without fractions: \( 18x = 36 \).

By cross-multiplying, the problem instantly becomes a straightforward linear equation, making it much easier to solve.

Solving Equations

Once you have cross-multiplied, you're often left with an algebraic equation that needs solving, just like in our example: \( 18x = 36 \). Solving equations typically involves finding the value of the unknown, represented by a variable like \( x \), that makes the equation true.The goal is to isolate \( x \) on one side of the equation. Here's what you do:

• Since \( 18x = 36 \), to solve for \( x \), divide both sides by 18, the coefficient of \( x \).
• This division simplifies the equation to \( x = \frac{36}{18} \).

Keep in mind that the steps of solving equations often involve operations like addition, subtraction, multiplication, and division. They're used strategically to simplify the equation and solve for the variable.

Simplifying Fractions

Fractions can often appear daunting, but simplifying them can make mathematics much more approachable. In the exercise, after solving the equation, you get \( x = \frac{36}{18} \). Now, it's time to simplify this fraction to its lowest terms.When simplifying fractions:

• Find the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 36 and 18 is 18.
• Divide both the numerator and the denominator by this GCD. So, divide 36 by 18 and 18 by 18.
• This calculation yields \( \frac{36}{18} = 2 \).

By reducing fractions to their simplest form, calculations become easier, and the results are clearer. Simplification is a crucial skill for making sense of numbers efficiently.