Problem 8 - Prealgebra

Question

Solve each proportion. $$\frac{4}{10}=\frac{8}{a}$$

Step-by-Step Solution

Answer

The value of \( a \) is 20.

Step 1: Understand the Proportion

We are given a proportion equation \( \frac{4}{10} = \frac{8}{a} \). This means that the ratio of 4 to 10 is equal to the ratio of 8 to some number \( a \). Our goal is to find the value of \( a \) that makes this equation true.

Step 2: Cross-Multiply

To solve for \( a \), we cross-multiply. This involves multiplying the numerator of each ratio by the denominator of the other ratio: \( 4 \times a = 10 \times 8 \). This gives us the equation \( 4a = 80 \).

Step 3: Solve for \( a \)

Now, solve for \( a \) by isolating it on one side of the equation. We do this by dividing both sides of the equation by 4: \( a = \frac{80}{4} \).

Step 4: Simplify the Solution

Simplify the equation \( a = \frac{80}{4} \) by performing the division. This results in \( a = 20 \).

Key Concepts

Understanding Ratios

A ratio is a simple way to compare two quantities. The numbers in the ratio represent how much of one thing there is compared to another. For example, a 4:10 ratio tells us that for every 4 units of one item, there are 10 units of another.

Think of ratios as a way to measure relationships. They show how two numbers are connected in size. This concept is used in various real-world scenarios, such as cooking where a recipe might require a 2:3 ratio of sugar to flour.

Ratios can be written in different forms, such as fractions. The fraction \( \frac{4}{10} \) represents the ratio of 4 to 10. This way of representing ratios makes them very useful in solving mathematical problems by allowing us to use techniques such as cross-multiplication.

Using Cross-Multiplication

Cross-multiplication is a powerful and simple method used to solve equations involving proportions. When you have two fractions set equal to each other, you can use this method to find an unknown number, like solving \( \frac{4}{10} = \frac{8}{a} \).

To cross-multiply, you multiply across the equal sign in a crisscross way:

• Multiply the numerator (top number) of the first fraction by the denominator (bottom number) of the second fraction: \(4 \times a\).
• Multiply the numerator of the second fraction by the denominator of the first fraction: \(10 \times 8\).

This gives you a new equation: \(4a = 80\).

Cross-multiplication works because it maintains the balance of the equation, allowing you to isolate and solve for the unknown variable.

Solving Equations with Variables

Once you have used cross-multiplication, you are left with a simple algebraic equation to solve. Consider the equation \(4a = 80\). The goal is to find the value of the variable \(a\) that makes the statement true.

Here’s how you tackle it:

• Look at the equation: \(4a = 80\). This tells you that 4 times some number \(a\) equals 80.
• To isolate \(a\), you need \(a\) by itself on one side of the equation. You achieve this by performing the opposite operation (division) to both sides.

Divide both sides of the equation by 4: \(a = \frac{80}{4}\). Doing the math, you find \(a = 20\).

This method shows how rearranging and simplifying an equation helps you find solutions efficiently, a key skill in solving algebra problems.