Problem 13 - 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. $$\frac{x}{4}=\frac{3}{8}$$

Step-by-Step Solution

Answer

The missing term x is \( \frac{3}{2} \).

Step 1: Understand the Proportion

The given proportion is \( \frac{x}{4} = \frac{3}{8} \). This equation states that the ratio of \(x\) to 4 is the same as the ratio of 3 to 8. We need to find the value of \(x\) that makes this statement true.

Step 2: Cross-Multiply

To find the missing term \(x\), we use the cross-multiplication property of proportions. This involves multiplying the numerator of one fraction by the denominator of the other:\[x \times 8 = 4 \times 3\]

Step 3: Solve the Equation

Now solve the equation obtained from cross-multiplication:\[8x = 12\]To solve for \(x\), divide both sides of the equation by 8:\[x = \frac{12}{8}\]

Step 4: Simplify the Fraction

The fraction \( \frac{12}{8} \) can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 4. Divide 12 by 4 to get 3, and divide 8 by 4 to get 2, thus simplifying the fraction:\[x = \frac{3}{2}\]

Key Concepts

Understanding Fractions

Fractions represent a part of a whole. They consist of two numbers, a numerator, and a denominator. The numerator is the top number, which indicates how many parts we have. The denominator is the bottom number, which shows how many equal parts the whole is divided into.
For example, in the fraction \( \frac{1}{2} \), 1 is the numerator, and 2 is the denominator. This fraction means that we have one out of two equal parts of something.
Fractions can be proper (numerator is less than the denominator), improper (numerator is greater than or equal to the denominator), or mixed numbers (a whole number and a fraction combined).
Understanding fractions is crucial for solving proportions, as these ratios are expressed as fractions. Grasping the role of each part of a fraction helps in setting up equations correctly.

Cross-Multiplication Made Easy

Cross-multiplication is a technique used to solve equations where two fractions are equal, known as proportions. For a proportion \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying each diagonal pair of terms. This gives us the equation \( a \times d = b \times c \).
This method is particularly useful because it transforms the fraction equation into a single equation without fractions. By doing so, it simplifies the process of solving for the unknown variable. Let's use an example from our exercise:

• Given: \( \frac{x}{4} = \frac{3}{8} \)
• Cross-multiply: \( x \times 8 = 4 \times 3 \)

The result is a simple linear equation that can be solved easily, leading to the solution for \( x \). Cross-multiplication is a reliable and efficient way to solve proportion problems.

The Art of Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. This involves dividing the numerator and denominator by their greatest common divisor (GCD).
To find the GCD, list the factors of both numbers and identify the largest factor they share. For instance, in the fraction \( \frac{12}{8} \), the factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 8 are 1, 2, 4, and 8. The greatest common factor is 4.

• Divide the numerator and denominator by 4: \( \frac{12}{8} = \frac{3}{2} \)

Simplifying makes fractions easier to understand and work with, ensuring they are reduced to their fundamental form. This is the final step in solving a proportion, confirming that the solution is accurate and simplified.