Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(x: 1=0.5: 5\)

Short answer

The value of \(x\) is 0.1.

Step-by-step solution

Understand the Proportion

The given proportion is \( x: 1 = 0.5: 5 \). This means that the ratios \( \frac{x}{1} \) and \( \frac{0.5}{5} \) are equal.

Set Up the Equation

The statement \( x:1 = 0.5:5 \) can be rewritten as an equation: \( \frac{x}{1} = \frac{0.5}{5} \).

Simplify the Right Side

Calculate \( \frac{0.5}{5} \): Divide 0.5 by 5 to get 0.1. So the equation becomes \( \frac{x}{1} = 0.1 \).

Solve for \(x\)

Since \( \frac{x}{1} = x \), the equation simplifies to \( x = 0.1 \).

Key concepts

Understanding Equivalent Ratios

In mathematics, a proportion showcases two equivalent ratios. A ratio is merely a comparison between two numbers. For example, the expression \( x:1 = 0.5:5 \) illustrates two ratios that are equal. Understanding these allows us to set up and solve equations.

To determine if two ratios are equivalent, consider the simplified form of each fraction.

• If we have \( \frac{x}{1} \) and \( \frac{0.5}{5} \), ensuring these are equal establishes a proportion.
• The essence of an equivalent ratio is that any multiplication or division applied to one ratio must be reciprocated in the other.

When ratios are equivalent, it means the two quantities change in direct proportion to each other. Thus, if one quantity increases, the other must increase in order to maintain the equality of the ratios.

Solving Equations in Proportions

Solving equations in a proportion primarily involves finding the unknown variable that maintains the balance between the two ratios. Let's pretend you have a proportion like \( x:1 = 0.5:5 \). You can express this proportion as an equation, written as \( \frac{x}{1} = \frac{0.5}{5} \).

To solve, follow these steps:

• **Convert Ratios to Equation:** Start by translating the ratio into a fraction equation. Here, it becomes \( \frac{x}{1} = \frac{0.5}{5} \).
• **Simplify the Fractions:** Perform the division on the numeric ratio. Calculate \( \frac{0.5}{5} \) to get 0.1.
• **Equalize the Equation:** Recognize that \( \frac{x}{1} \) simplifies to \( x \), making it \( x = 0.1 \).

Solving such equations allows for determining the value of unknowns using straightforward arithmetic. Thus, one key step in solving is ensuring any fraction reduction is done accurately.

Mastering Division with Decimals

When working with proportions and solving equations, dividing with decimals becomes a frequent necessity. For instance, to solve the given equation, \( \frac{0.5}{5} \), division is required. Here's how it works.

• **Set Up the Division:** As in solving \( \frac{0.5}{5} \), align 0.5 under the division bar with 5 as the divisor.
• **Perform the Division:** Divide as you would whole numbers. Since 5 goes into 0.5 as 10 goes into 1, place the decimal correctly resulting in 0.1.
• **Precision in Calculations:** Be diligent about placing the decimal correctly after performing the division. Round only when necessary, sticking to specified decimal places – often two.

Understanding this method is crucial since it forms a basis for solving numerous mathematical problems. Once you're confident, division with decimals becomes a straightforward task, essential for topics involving proportions and more complex equations in mathematics.