Question

Find the value of \(x.\) $$4: 6=x: 4$$

Short answer

\(x = \frac{8}{3}\)

Step-by-step solution

Identify the Type of Problem

Recognize that the given equation is a proportion, which can be solved by cross-multiplication. A proportion is an equation that states that two ratios are equivalent.

Cross-Multiply

Cross-multiply the terms of the proportion to create an equation. Multiply 4 by 4 and 6 by x. This yields the equation: $$4 \times 4 = 6 \times x.$$

Perform the Multiplication

Perform the multiplication on both sides of the equation to simplify it: $$16 = 6x.$$

Solve for x

Divide both sides of the equation by 6 to solve for x: $$\frac{16}{6} = \frac{6x}{6} \implies x = \frac{8}{3}.$$ Therefore, the value of x is \(\frac{8}{3}\) or approximately 2.67.

Key concepts

Cross-Multiplication

When faced with a proportion problem, cross-multiplication is a powerful strategy that transforms ratios into an easier-to-solve algebraic equation. It involves multiplying across the 'equals' sign to eliminate the fractions. For example, if we have a proportion like \(a:b = c:d\), cross-multiplication involves multiplying \(a\) by \(d\) and \(b\) by \(c\), yielding \(a \times d = b \times c\).

Here's why cross-multiplication works: it's based on the property that if two ratios are equal, their cross products are also equal. This principle holds true for any true proportion and provides a simplified method for solving for an unknown variable. In our exercise \(4:6 = x:4\), by cross-multiplying, we create a scenario where the variable \(x\) stands alone on one side of the equation, making it straightforward to solve.

Ratio and Proportion

A ratio is a way to compare two quantities by using division as in \(a:b\), where \(b\) is not zero. When two ratios are set equal to each other, they form a proportion, such as \(a:b = c:d\), indicating that the two ratios are equivalent.

Understanding proportions is vital as they are used to represent the same relationship between numbers, regardless of their scale. That's why proportions can model real-world situations like maps, recipes, or scaling objects. Proportions tell us that if one quantity changes, the other changes in a way that the ratio remains constant. To solve proportions like in the exercise, we can utilize methods such as cross-multiplication to find the value of an unknown variable.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. When an equation contains an unknown variable, the goal is often to solve for that variable. This process involves performing operations that will isolate the variable on one side of the equation.

In the context of proportions, the created algebraic equation after cross-multiplication allows us to solve for the unknown by isolating it. For instance, after converting our ratio into an equation \(16 = 6x\), we can divide both sides by 6 to find \(x\), thereby applying algebraic principles to solve the proportion problem we were presented with. Algebraic equations are a foundational concept in math, and understanding how to manipulate and solve them is crucial for tackling a wide range of mathematical problems.