Problem 15 - Prealgebra

Question

Find the missing term in each of the following proportions. Set up each problem like the examples in this section. Write your answers as fractions in lowest terms. $$\frac{5}{9}=\frac{x}{2}$$

Step-by-Step Solution

Answer

The missing term is \( x = \frac{10}{9} \).

Step 1: Set up the Equation

We are given the proportion \( \frac{5}{9} = \frac{x}{2} \). The goal is to find the value of \( x \). We will use cross-multiplication to solve this equation.

Step 2: Cross-Multiply

Apply cross-multiplication to the equation. This means we multiply the numerator of the first fraction by the denominator of the second fraction and the numerator of the second fraction by the denominator of the first fraction, which gives us: \( 5 \times 2 = 9 \times x \).

Step 3: Simplify the Equation

Simplifying the cross-multiplied equation gives us \( 10 = 9x \).

Step 4: Solve for x

To find \( x \), divide both sides of the equation by 9: \( x = \frac{10}{9} \).

Step 5: Verify the Solution

Double-check the result. Substitute \( x = \frac{10}{9} \) back into the original proportion to ensure equality: \( \frac{5}{9} = \frac{\frac{10}{9}}{2} \), which simplifies to \( \frac{5}{9} = \frac{5}{9} \). Therefore, our solution is correct.

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a powerful technique used to solve proportions, making it simple to find unknown variables when fractions are involved. Here, we are dealing with two fractions set equal to each other, which is known as a proportion. The methodology works by eliminating the fractions and allowing us to work with whole numbers instead. This is done through the following steps:

• First, identify the numerators and denominators of each fraction.
• Then, multiply the numerator of the first fraction by the denominator of the second fraction.
• Simultaneously, multiply the numerator of the second fraction by the denominator of the first fraction.

For instance, if you have the proportion \( \frac{5}{9} = \frac{x}{2} \), cross-multiplication would lead to the equation \( 5 \times 2 = 9 \times x \). This effectively gets rid of the fractions, paving the way for straightforward algebraic manipulation.

Solving Equations the Simplified Way

Once you have executed cross-multiplication, you're often left with an algebraic equation where only one variable is unknown. Solving these equations requires isolating the variable, making it the subject of the equation. Here’s a straightforward way to do it:

• First, simplify the equation obtained from cross-multiplication. For our example, we start with \( 10 = 9x \).
• Reorganize the equation to isolate \( x \). In this case, divide both sides by 9 to get \( x = \frac{10}{9} \).
• This process transforms the problem from a proportion question into a simple division, which is much easier to solve.

By carefully following these steps, solving equations after cross-multiplication becomes an easy task.

A Look into Fractions

Dealing with fractions is a fundamental part of mathematics, especially when solving proportions. Fractions represent a part of a whole and can sometimes be daunting due to their dual-component nature—numerators and denominators. When working with proportions such as \( \frac{5}{9} = \frac{x}{2} \), fractions must be manipulated until they reach their lowest terms:

• Keep an eye on simplification once you've cross-multiplied and solved for the variable. Always reduce fractions to their simplest form.
• To reduce a fraction, find the greatest common divisor (GCD) of the numerator and denominator. Divide both by this number.
• In our example, \( \frac{10}{9} \) is already in lowest terms since 10 and 9 have no common divisors other than 1.

Understanding and managing fractions skillfully is crucial for efficiently solving mathematical problems involving proportions.