Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(12: 1=x: 5.5\)

Short answer

The solution is \(x = 66\).

Step-by-step solution

Identify the Proportion

The problem requires solving for \(x\) where the proportion given is \(\frac{12}{1} = \frac{x}{5.5}\). This means that the ratios 12 to 1 is equal to \(x\) to 5.5.

Set Up the Equation

Using the concept of cross-multiplication for proportion, we have the equation \(12 \times 5.5 = 1 \times x\). This sets up the problem to find \(x\).

Perform Cross-Multiplication

Perform the multiplication on the left-hand side: \(12 \times 5.5 = 66\). This gives us \(66 = x\).

Solve for x

From the equation \(66 = x\), it is clear that \(x = 66\).

Check the Solution

We can verify by substituting \(x = 66\) back into the original proportion: \(\frac{12}{1} = \frac{66}{5.5}\). Simplifying the right-hand side gives \(12 = 12\), confirming that \(x = 66\) is correct.

Key concepts

Understanding Cross-Multiplication

Cross-multiplication is a powerful tool that arises when dealing with proportions. In a proportion like \(\frac{a}{b} = \frac{c}{d}\), the cross-multiplication principle tells us that \(a \times d = b \times c\). This is extremely helpful for solving unknowns in a proportion form. By cross-multiplying, we eliminate the fractions and form a simple equation that can be more manageable.

• This technique works because of the property of equality, maintaining the balance of the equation.
• It helps simplify complex ratios into simpler forms, making calculations straightforward.

Try using cross-multiplication whenever you're faced with proportions to make your life easier, especially when hunting for unknowns!

Decoding Ratios

Ratios are essential for expressing how many times one quantity is contained within another. They can be written as fractions, using the colon symbol (12:1), or with the word 'to' (12 to 1). It's like making a comparison between two numbers, simplifying them down to their basic relationship.

• A ratio can tell you how often something occurs.
• It is a foundational concept used in mathematics to explain relationships between numbers.

Understanding ratios is crucial because they provide a stepping stone to solving proportions and similar mathematical problems.

Approaching Solving Equations

Solving equations effectively is a crucial skill in mathematics. An equation is a mathematical statement that asserts the equality of two expressions. To solve, your goal is to find the value of the unknown variable that satisfies this statement.

• Start by isolating the variable. Try to move everything else to the other side of the equation.
• Methods like cross-multiplication simplify the process when dealing with proportions.

Always verify your solution by substituting the value back into the equation to see if it holds true. This ensures that your solution is not only theoretically correct but practically viable.

Emphasizing the Importance of Mathematics Education

Mathematics education is a cornerstone of a functioning society. It equips individuals with problem-solving skills and the ability to think logically and systematically. Understanding basic concepts like ratios and equations lays the groundwork for more advanced mathematical topics.

• It helps in developing critical thinking skills necessary for various real-life scenarios.
• Engaging with mathematics enhances cognitive abilities and sharpens the mind.

Emphasizing a strong foundation in mathematics from an early age can significantly impact one's ability to analyze and interpret data, making informed decisions in everyday life.