Problem 78 - Prealgebra

Question

Solve each proportion. $$\frac{25}{4}=\frac{x}{100}$$

Step-by-Step Solution

Answer

\( x = 625 \)

Step 1: Understand the Proportion

A proportion states that two ratios or fractions are equal. Here, we have \( \frac{25}{4} = \frac{x}{100} \). Our goal is to solve for \( x \).

Step 2: Cross-Multiply

To solve for \( x \) in a proportion, we can use cross-multiplication. This means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa:\[25 \times 100 = 4 \times x\]

Step 3: Perform the Multiplication

Calculate the products from cross-multiplication:\[2500 = 4x\]

Step 4: Solve for x

To isolate \( x \), divide both sides of the equation by 4:\[x = \frac{2500}{4}\]

Step 5: Simplify the Division

Perform the division to find \( x \):\[x = 625\]

Key Concepts

Cross-Multiplication

Cross-multiplication is a fundamental technique used for solving proportions. It essentially helps us eliminate fractions by transforming a proportion into a simple equation, allowing for easier manipulation.

When facing a problem like \( \frac{25}{4} = \frac{x}{100} \), cross-multiplication involves multiplying diagonally: the numerator from one side with the denominator of the other side. Here's a step-by-step breakdown:

• Multiply 25 (numerator of the first fraction) by 100 (denominator of the second fraction), which gives us 2500.
• Multiply 4 (denominator of the first fraction) by \( x \) (numerator of the second fraction), giving 4\( x \).

From here, you'll have a simplified equation: \( 2500 = 4x \). The fractions are gone, and you're left with a straightforward equation to solve for \( x \). This powerful yet simple method makes dealing with proportions quick and efficient.

Solving Equations

Once you've cross-multiplied, the next step is solving an equation, which usually involves isolating the variable to one side. For the equation \( 2500 = 4x \), we can isolate \( x \) by dividing both sides of the equation by 4.

Let's break it down:

• Take the entire equation: \( 2500 = 4x \).
• Divide both sides by 4 to get \( x \) alone on one side: \( x = \frac{2500}{4} \).

This division isolates \( x \) and allows us to determine its value. As with any equation, ensuring that calculations are done correctly at each step is crucial. Performing accurate arithmetic will yield \( x = 625 \), the solution for this particular problem. Remember, solving equations is like unwrapping a present; you gently unwrap the layers (in this case, arithmetic steps) to reveal the solution.

Ratios and Fractions

Understanding the relationship between ratios and fractions is key to mastering proportions. A ratio compares two quantities, while a fraction represents parts of a whole. In the context of proportions, both express the same fundamental idea.

Let's consider the proportion \( \frac{25}{4} = \frac{x}{100} \) again. Here, \( \frac{25}{4} \) represents a comparison of the number 25 to 4, and \( \frac{x}{100} \) compares \( x \) to 100.

When two ratios are equal, as in a proportion, it means the relationship between the numbers on one side matches the relationship on the other side. This equality allows us to solve for unknowns using methods like cross-multiplication.

• Fractions like \( \frac{25}{4} \) show how one part relates to another part.
• Ratios represent the same idea as fractions but often compare different units or entities.

Sometimes, these small distinctions can help clarify how we understand and work with mathematical concepts. Learning to see and handle ratios and fractions interchangeably will deepen your comprehension and make problems involving proportions much more manageable.