Problem 4 - Algebra A Combined Function

Question

Solve each proportion. $$ \frac{9}{4 x}=\frac{6}{2} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{3}{4} \).

Step 1: Understand the Proportion

A proportion is an equation that states that two ratios are equal. The given equation is \( \frac{9}{4x} = \frac{6}{2} \), meaning the ratio of 9 to \( 4x \) is equal to the ratio of 6 to 2.

Step 2: Cross Multiply

To solve the proportion, use cross-multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction and vice versa: \( 9 \times 2 = 6 \times 4x \).

Step 3: Simplify the Equation

Perform the multiplications: 9 * 2 = 18 and 6 * 4x = 24x, giving us the equation \( 18 = 24x \).

Step 4: Solve for x

To find the value of \( x \), divide both sides of the equation by 24:\( \frac{18}{24} = x \).

Step 5: Simplify the Fraction

Simplify the fraction \( \frac{18}{24} \) by finding the greatest common divisor (GCD) of 18 and 24, which is 6. Then, divide both numerator and denominator by 6:\( \frac{18}{24} = \frac{3}{4} \).

Step 6: Conclusion

The value of \( x \) that solves the proportion is \( \frac{3}{4} \).

Key Concepts

Cross Multiplication Made Simple

Cross-multiplication is a straightforward technique used to solve proportions, which are equations asserting that two ratios are equal. In the given problem, the proportion is \( \frac{9}{4x} = \frac{6}{2} \). To apply cross-multiplication, follow these easy steps:

• Identify the numerators and denominators of both fractions.
• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the numerator of the second fraction by the denominator of the first fraction.

Here’s how it looks with the actual numbers from the exercise: \( 9 \times 2 \) and \( 6 \times 4x \). These operations yield the products \( 18 \) and \( 24x \). Placing them equal because of the proportion gives us the equation \( 18 = 24x \), which sets the stage for solving for \( x \). This technique is powerful because it eliminates fractions and leads directly to a solvable equation.

Effortless Simplification of Fractions

Simplifying fractions is an essential skill that helps with simplifying equations. It involves reducing a fraction to its smallest form by removing common factors shared by the numerator and the denominator. Take the fraction \( \frac{18}{24} \) from our example. Begin by finding the greatest common divisor (GCD) of 18 and 24.
In this case, the GCD is 6. Now, divide both the numerator and denominator of the fraction by the GCD:

• When you divide 18 by 6, you get 3.
• When you divide 24 by 6, you get 4.

As a result, \( \frac{18}{24} = \frac{3}{4} \), meaning that the simplified solution for \( x \) in the proportion is \( \frac{3}{4} \). By simplifying fractions, calculations become more straightforward, and you gain a clearer understanding of the relationships in proportion problems.

Understanding Ratios in Proportions

Ratios are relationships between two numbers expressing how many times the first number contains the second or how they compare in size. In the context of proportions, these ratios are shown as fractions. The original problem \( \frac{9}{4x} = \frac{6}{2} \) presents two such ratios.
When both ratios are equal, as indicated by the equal sign, a proportion is formed. By recognizing the presence of equal ratios, you understand that the comparison between the numbers remains consistent. This understanding is crucial when employing cross-multiplication, as it ensures the relationships are balanced after eliminating the fractions.
Exploring ratios also helps in identifying proportional relationships in real-world scenarios, such as scaling recipes or adjusting quantities. Understanding the concept of ratios is not only instrumental in solving such textbook exercises but in practical decision-making as well.

Solving Equations to Find Unknowns

Solving equations is about finding the value of unknown variables. In proportion problems like the one in this exercise, after cross-multiplication, you end up with an equation in a more workable format. Here, the equation \( 18 = 24x \) needs to be solved.

• Start by isolating the variable \( x \).
• Do this by performing the same operation on both sides. In this exercise, divide both sides by 24 to solve for \( x \).

This results in \( x = \frac{18}{24} \), which can then be simplified to \( \frac{3}{4} \) by simplifying fractions.
By breaking down equations this way, you systematically discover the unknowns. Mastering equation solving is a foundation for algebra and beyond, proving invaluable as you approach increasingly complex mathematical scenarios.