Problem 52 - Prealgebra

Question

Solve each equation. $$\frac{x}{3.5}=\frac{85}{1}$$

Step-by-Step Solution

Answer

The solution is \( x = 297.5 \).

Step 1: Understand the Equation

The given equation is \( \frac{x}{3.5} = \frac{85}{1} \). This is a proportional relationship, meaning the two fractions are equal. Our task is to solve for \( x \).

Step 2: Cross Multiply to Eliminate the Denominator

Cross-multiply to convert the equation into a simpler form without fractions. Multiply both sides by their respective denominators: \( x \times 1 = 85 \times 3.5 \). This simplifies to \( x = 85 \times 3.5 \).

Step 3: Perform the Multiplication

Multiply 85 by 3.5 to find the value of \( x \). Calculate \( 85 \times 3.5 = 297.5 \).

Step 4: Verify the Solution

Substitute \( x = 297.5 \) back into the original equation to verify. The left-hand side becomes \( \frac{297.5}{3.5} \), which simplifies to 85, matching the right-hand side. Therefore, the solution is correct.

Key Concepts

Understanding Cross Multiplication

Cross multiplication is a handy technique used to simplify equations involving fractions. It helps eliminate the fractions by multiplying across the denominators and numerators. In our example, the equation \( \frac{x}{3.5} = \frac{85}{1} \) can be cross-multiplied to get rid of the denominators.
By cross-multiplying, we multiply the numerator of each fraction by the denominator of the other:

• Multiply \( x \) by 1 (the denominator of the second fraction): \( x \times 1 \).
• Multiply 85 by 3.5 (the denominator of the first fraction): \( 85 \times 3.5 \).

This process leaves us with the simple equation \( x \times 1 = 85 \times 3.5 \). Cross multiplication transforms a complex fraction-based equation into an easy multiplication problem, making it simpler to solve. In most cases, it converts a two-step operation into a straightforward calculation.

Exploring Proportional Relationships

Proportional relationships are a fundamental concept in math, where two quantities increase or decrease at the same rate. They can be expressed as equations in the form of fractions set equal to each other, like \( \frac{x}{3.5} = \frac{85}{1} \).
When two ratios are proportional, it means that multiplying or dividing each part of the ratio by the same number results in equivalent expressions. In the given exercise, the right-hand side ratio \( \frac{85}{1} \) of the equation shows that for every 1 unit in the denominator, there are 85 units in the numerator.

• This proportionality allows us to find unknown values through operations such as cross-multiplication.
• It helps maintain the balance of both sides of an equation, essential for solving for variables.

Understanding proportional relationships isn't just critical for equations, but also applies to real-world scenarios like scaling recipes, creating maps, or analyzing data trends.

Verifying Solutions for Accuracy

Verifying solutions is the final crucial step in solving equations. After calculating the value of \( x \), it’s essential to check that this solution satisfies the original equation. In the given scenario, we found \( x = 297.5 \), but how can we be sure this is the right answer?
To verify, substitute this value of \( x \) back into the original equation:

• Replace \( x \) with 297.5, giving \( \frac{297.5}{3.5} \).
• Calculate the division: \( 297.5 \div 3.5 = 85 \).

Since 85 equals the other side of the equation, \( \frac{85}{1} \), the solution is verified as correct. This verification process ensures there are no errors in calculation or logic, reinforcing the accuracy and reliability of the solution found. Always check your work by substituting back to confirm the correctness of your results.