Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(15: 20=x: 30\)

Short answer

x = 22.5

Step-by-step solution

Understand Proportions

A proportion compares two ratios or fractions. In our problem, we have the proportion \( \frac{15}{20} = \frac{x}{30} \). Our goal is to solve for \( x \).

Set the Cross-Multiplication Equation

Cross-multiplication is a method to solve proportions. Multiply the numerator of each ratio by the denominator of the other: \( 15 \times 30 = 20 \times x \).

Simplify the Multiplications

First, calculate \( 15 \times 30 = 450 \). The equation becomes \( 450 = 20x \).

Solve for x

To isolate \( x \), divide both sides by 20: \( x = \frac{450}{20} \).

Perform the Division

Divide 450 by 20. The division gives \( x = 22.5 \).

Key concepts

Cross-Multiplication

Cross-multiplication is a powerful technique for solving problems involving proportions. When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying the numerator of one ratio by the denominator of the other ratio. This method works because it creates a simple equation that can be easily solved.

In the provided example, the proportion is \( \frac{15}{20} = \frac{x}{30} \). By cross-multiplying, we create the equation \( 15 \times 30 = 20 \times x \). This is done by multiplying the numerator of the first ratio (15) with the denominator of the second ratio (30), and vice versa.

• Step 1: Multiply 15 and 30
• Step 2: Multiply 20 and \( x \)

You end up with an equation that can be solved easily for \( x \). This process is straightforward and avoids the need for more complex algebra.

Ratios

A ratio is a comparison between two numbers, showing how many times one value contains or is contained within the other. Ratios can be written in different forms like 3:4, \( \frac{3}{4} \), or 3 to 4. In the context of proportions, ratios are used to represent relationships between quantities.

The problem we are solving involves the ratio \( 15:20 \) compared to \( x:30 \). Here, the two ratios are shown in a proportional relationship, meaning they are equal and can be written as \( \frac{15}{20} = \frac{x}{30} \). This sets the stage for using cross-multiplication to find the unknown value.

• Understanding the given ratios is key to setting up the correct proportion.
• Make sure to align the quantities correctly when setting up your problem.

By recognizing these ratios as equal, it becomes easy to apply further problem-solving methods.

Equations

In mathematics, an equation is a statement that shows the equality of two expressions. Solving proportions typically leads to forming an equation, as seen in the previous examples. The cross-multiplication technique leads directly to setting up such equations.

For the given proportion \( \frac{15}{20} = \frac{x}{30} \), cross-multiplying results in the equation \( 15 \times 30 = 20 \times x \), which simplifies to \( 450 = 20x \). Here, we have structured an equation typical in algebra that allows us to solve for the unknown.

• Understand that forming an equation is crucial for solving any algebraic problem.
• Be precise with each step to ensure the equation is correctly set up.

Once the equation is correctly formed, the next step is to solve it, ensuring an accurate solution for \( x \).

Problem Solving

Problem-solving in mathematics involves a systematic approach to finding a solution to a given problem. When working with proportions, understanding each component and their relationships are key.

The exercise demonstrates a clear path to solving for \( x \) in the given proportion. It starts with setting up the proportion equation, applying cross-multiplication, and simplifying to find \( x \). Each step requires attention to detail and accuracy.

• Break down the problem into smaller steps.
• Verify each step for potential calculation errors.
• Practice these steps to gain confidence in problem-solving.

By following a methodical approach, like the step-by-step guide provided, problem-solving becomes easier and more manageable, leading to a clearer understanding of mathematical concepts.