Question

\(\frac{2}{3}=\frac{x}{9}\)

Short answer

The solution to the proportion \(\frac{2}{3}=\frac{x}{9}\) is \(x = 6\).

Step-by-step solution

Identify the proportional elements

In this proportion \(\frac{2}{3}=\frac{x}{9}\), \(2\) and \(x\) are numerators, \(3\) and \(9\) are denominators.

Cross-Multiply

Cross multiplication means multiplying the numerator of one of the ratios by the denominator of the other ratio, and vice versa. So here, \(2*9 = 18\) and \(x*3=3x\). Set the two products equal to each other to get \(18 = 3x\).

Solve for x

To solve for \(x\), divide both sides of the equation \(18 = 3x\) by \(3\). This gives \(x = \frac{18}{3}\).

Simplify the result

Simplifying the fraction obtained in Step 3 gives \(x = 6\).

Key concepts

Cross-Multiplication

Cross-multiplication is a common method used in solving algebraic proportions, like the equation \(\frac{2}{3}=\frac{x}{9}\). When you cross-multiply, you are essentially connecting the two ratios by creating an equation that can be simplified and solved.

In the given proportion, we start by multiplying the numerator of one fraction by the denominator of another fraction. Specifically, we calculate the product of \(2\) and \(9\) to get \(18\), and the product of \(x\) and \(3\) to get \(3x\).

By setting these two products equal to each other, you have effectively transformed the fractions into an equation \(18 = 3x\). This approach helps eliminate the fractions, making it easier to solve for the unknown variable. Cross-multiplication is a helpful method when dealing with equal ratios, as it keeps the relationship between terms intact while transforming the problem into a more manageable form.

Solving Equations

Once you have your basic equation from cross-multiplying, the next task is to isolate the variable. This process is straightforward and only involves basic arithmetic operations.

In the equation \(18 = 3x\), our aim is to solve for \(x\) by getting it alone on one side of the equation. To achieve this, you need to 'cancel out' the coefficient attached to \(x\).

• First, divide both sides of the equation by \(3\) to isolate \(x\).
• This gives us \(x = \frac{18}{3}\).

By performing the division \(x = \frac{18}{3}\), we simplify the equation to obtain \(x = 6\). Solving equations often involves basic operations such as addition, subtraction, multiplication, or division. Simplifying each step helps to clearly see the progression and eventually reach the solution.

Fraction Simplification

Fraction simplification is an essential skill in algebra that makes numbers easier to work with and understand. After solving the core equation \(18 = 3x\), we derived \(x = \frac{18}{3}\). At this point, simplifying is necessary to find a more concise expression for \(x\).

Simplification involves reducing the fraction to its simplest form, by dividing the numerator and the denominator by their greatest common divisor (GCD). Here, both \(18\) and \(3\) are divisible by \(3\).

• Divide \(18\) by \(3\) to get \(6\).
• Similarly, divide \(3\) by \(3\) to get \(1\).

Hence, the simplified form of \(\frac{18}{3}\) is \(6\). Always remember that the goal of fraction simplification is to express the solution as tidily and accurately as possible, making it easier to interpret in relation to the problem at hand. By reaching \(x = 6\), not only have we found the solution, but we have also presented it in its most user-friendly form.