Problem 3 - Beginning and Intermediate Algebra

Question

Solve each proportion. See Examples 1 and \(2 .\) $$ \frac{x}{10}=\frac{5}{9} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{50}{9}\).

Step 1: Set up the cross multiplication

To solve the proportion \(\frac{x}{10}=\frac{5}{9}\), we start by cross-multiplying the terms. This involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa.

Step 2: Perform the cross multiplication

We multiply \(x\) by \(9\) and \(5\) by \(10\) to get:\[x \times 9 = 5 \times 10\]

Step 3: Simplify the equation

Calculate the right side of the equation from Step 2:\[9x = 50\]

Step 4: Solve for \(x\)

To find \(x\), divide both sides of the equation by 9:\[x = \frac{50}{9}\]

Step 5: Verify the solution

Check that substituting \(x = \frac{50}{9}\) back into the original proportion shows equality:\[\frac{50/9}{10} = \frac{5}{9}\]Simplifying \(\frac{50/9}{10}\) gives \(\frac{5}{9}\), confirming the solution is correct.

Key Concepts

Understanding Cross Multiplication

When you encounter problems involving proportions like \(\frac{x}{10} = \frac{5}{9}\), cross multiplication is a powerful technique to solve them. It transforms the problem into a simple equation that we can easily handle. But what does cross multiplication mean?

• Multiplication Across the Equal Sign: When you multiply, you take the numerator of one fraction and multiply it with the denominator of the other fraction across the equal sign. At the same time, do this in the reverse direction as well.
• Example in Practice: For \(\frac{x}{10} = \frac{5}{9}\), you multiply \(x\) by \(9\) and \(5\) by \(10\). This gives you the equation \(9x = 50\).
• A Useful Tool for Proportions: Cross multiplication helps because it clears fractions and directly leads to solving for the unknown variable.

This technique is not just for solving; it's also a quick check for equality between two ratios.

Solving Equations Step by Step

After applying cross multiplication, you typically end up with a basic algebraic equation. Solving this involves getting the variable of interest alone on one side. Let's look at the steps closely:

• Isolate the Variable: From our cross multiplication result \(9x = 50\), you'll need to get \(x\) by itself. This often involves dividing both sides of the equation by the number in front of \(x\).
• Execute the Division: In this example, divide both sides by \(9\). You'll get \(x = \frac{50}{9}\).
• Checking Work: A crucial last step! Substitute \(x = \frac{50}{9}\) back into the original proportion to verify it holds true. This doesn’t just check the calculations; it ensures our understanding is correct.

Each problem might have different complexities, but these steps maintain their usefulness in handling linear equations derived from proportions.

Algebraic Fractions in Proportions

Algebraic fractions, like those found in proportion problems, can be lots of fun once you get the hang of them. They involve variables which can sometimes make things look more complicated than they really are.

• Why Algebraic? These fractions include variables (like \(x\) in \(\frac{x}{10}\)), which means we need to apply algebraic techniques to solve them.
• Simplicity through Operations: Handling algebraic fractions often requires simplifying through operations like cross multiplication or factoring. These operations make it easier to isolate and solve for the unknown.
• Be Clear on the Basics: Understanding what each part of the fraction does, and how operations affect the whole fraction, is key. Remember, the goal is to simplify and solve effectively.

When dealing with algebraic fractions in proportions, practice will reinforce the connections between mathematical operations and the outcome, making these problems less daunting over time.