Problem 36 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{27}{x}=\frac{9}{4} $$

Step-by-Step Solution

Answer

The value of \( x \) is 12.

Step 1: Understand the Proportion

A proportion is an equation that states that two ratios are equal. We are given the proportion \( \frac{27}{x} = \frac{9}{4} \). Our goal is to find the value of \( x \).

Step 2: Cross-Multiply

To solve for \( x \), use cross-multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction, and the denominator of the first fraction by the numerator of the second fraction: \[27 \times 4 = 9 \times x\]

Step 3: Simplify the Equation

After cross-multiplying, simplify the equation: \[108 = 9x\]This equation represents the relationship between the numbers after cross-multiplying.

Step 4: Solve for X

To isolate \( x \), divide both sides of the equation by 9:\[\frac{108}{9} = x\]Simplify the expression on the left side to find the value of \( x \):\[ x = 12 \]

Step 5: Verify the Solution

To ensure our solution is correct, substitute \( x = 12 \) back into the original proportion:\[\frac{27}{12} = \frac{9}{4}\]Simplify \( \frac{27}{12} \) to see if it equals \( \frac{9}{4} \):\[\frac{27 \div 3}{12 \div 3} = \frac{9}{4}\]This confirms our solution is correct.

Key Concepts

Cross-Multiplication in Proportions

When dealing with proportions, a common technique to solve for a variable is cross-multiplication. This method involves multiplying across the equals sign in a crisscross or diagonal fashion. Consider the proportion given in the exercise: \( \frac{27}{x} = \frac{9}{4} \). Here’s how cross-multiplication works in this scenario:

• Step 1: Identify the numerators and denominators. In this case, 27 and 9 are the numerators, while \(x\) and 4 are the denominators.
• Step 2: Multiply the numerator of one side by the denominator of the other side. That is, multiply 27 by 4 and \(x\) by 9.
• Step 3: Set these two products equal to each other. You get: \(27 \times 4 = 9 \times x\).

Now you have successfully created an equation that can be solved for the unknown variable \(x\). Cross-multiplication is a powerful and simple method to find unknowns in proportion problems.

Solving Equations from Proportions

Once you have used cross-multiplication to form an equation, the next step is to solve it. If we look at the equation \(27 \times 4 = 9 \times x\), it simplifies to \(108 = 9x\).

• First: Simplify both sides as much as possible. Here, our focus is to isolate \(x\).
• Second: Perform the necessary division. We divide both sides of the equation by 9 to solve for \(x\).
• Calculate: \( \frac{108}{9} = x \), thus \( x = 12 \).

By solving this equation, you isolate \(x\) and determine its value. This step-by-step approach is essential in handling proportions and finding unknown variables effectively.

Verifying Solutions to Proportions

It’s not enough to just find the solution; verifying your answer is crucial to ensure accuracy. In checking the solution for a proportion, you substitute the found value back into the original equation and see if the equality holds.

Here’s how the verification is performed for our original proportion \( \frac{27}{x} = \frac{9}{4} \), with \( x = 12 \):

• Replace \(x\): Substitute 12 in place of \(x\) to get \(\frac{27}{12}\).
• Simplify: Simplify \(\frac{27}{12}\) by dividing both the numerator and the denominator by their greatest common factor, which is 3, resulting in \(\frac{9}{4}\).
• Compare: \(\frac{9}{4}\) equals \(\frac{9}{4}\), confirming the calculations are correct.

Verifying solutions is an essential process that ensures confidence in your mathematical work, providing a reliable check for your problem-solving journey.