Problem 50 - Elementary Algebra Within Reach

Question

Solve the proportion. $$\frac{x}{3}=\frac{28}{12}$$

Step-by-Step Solution

Answer

The solution to the proportion is x = 7.

Step 1: Identify the proportions

The equation provided gives a relation between two fractions via a proportion, which is given by \(\frac{x}{3} = \frac{28}{12}\). Here 'x' is the variable to solve for.

Step 2: Cross-multiply

In this step, apply the property of proportions boundaries which states that the product of means equals the product of extremes. This involves cross multiplying the terms: multiply 'x' and '12' and '3' and '28'.

Step 3: Perform the multiplication

Perform the multiplication of the cross multiplied terms. This results in the equation 12x = 84.

Step 4: Solve for x

In order to find the value of 'x', divide both sides of the equation by 12. This gives the result x = 7

Key Concepts

Solving Proportions

Proportions are equations that state two ratios are equal. In the equation \( \frac{x}{3} = \frac{28}{12} \), we have two ratios: \( \frac{x}{3} \) and \( \frac{28}{12} \). Solving proportions involves finding a number that satisfies the equation by making the two ratios equivalent.

To solve a proportion, you need to balance these ratios by manipulating the algebraic equation provided. You need to

• Identify what 'x' represents in the context of the equation.
• Understand the structure of the equation; here, it equates two fractions or ratios.

Once you've identified the variable, the next step is to apply algebraic techniques, such as cross-multiplication, to find the correct value for 'x'.

Cross-Multiplication

Cross-multiplication is a method used to solve equations involving proportions. By using this technique, we eliminate the fractions, making it easier to solve for the variable.

In the proportion \( \frac{x}{3} = \frac{28}{12} \), cross-multiplication involves multiplying the numerator of each fraction by the denominator of the opposite fraction:

• Multiply \(x\) by 12.
• Multiply 3 by 28.

This results in the equation: \[12x = 3 \times 28\]This simplifies the original proportion into a linear equation without fractions, allowing us to proceed with solving for 'x'. It's important to ensure that the products on both sides of the equation remain equal, preserving the equality.

Variable Isolation

Variable isolation is the process of rearranging an equation to find the value of a variable. After cross-multiplying, our equation becomes \(12x = 84\). To isolate 'x', we need to get 'x' alone on one side of the equation.

This involves using inverse operations. In this case, since 'x' is multiplied by 12, we do the opposite operation—divide both sides of the equation by 12:\[\frac{12x}{12} = \frac{84}{12}\] This simplifies further to \( x = 7 \). Now 'x' is isolated on one side of the equation, revealing its value. Understanding variable isolation is crucial, as it transforms the equation and leads directly to the solution.