Problem 16 - 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. $$\frac{3}{7}=\frac{x}{3}$$

Step-by-Step Solution

Answer

The missing term is \( \frac{9}{7} \).

Step 1: Set Up the Proportion Equation

The given proportion is \( \frac{3}{7} = \frac{x}{3} \). This means that the two ratios are equal. We need to find the value of \( x \) that makes this equation true.

Step 2: Cross-Multiply

To solve for \( x \), cross-multiply the terms of the proportion. This involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. This gives us:\[ 3 \times 3 = 7 \times x \]

Step 3: Simplify the Equation

Now, simplify the expression obtained from cross-multiplication: \( 9 = 7x \).

Step 4: Solve for x

To find \( x \), divide both sides of the equation by 7: \[ x = \frac{9}{7} \]

Step 5: Ensure the Fraction is in Lowest Terms

The fraction \( \frac{9}{7} \) is already in its simplest form, as 9 and 7 have no common factors other than 1.

Key Concepts

Cross-Multiplication

Cross-multiplication is a handy method used to solve proportions. It allows you to find a missing value in a proportion by converting the fractions into a simpler equation. When dealing with a proportion, such as \(\frac{3}{7} = \frac{x}{3}\), cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction on the opposite side of the equation. Here’s how it works:

• Multiply the numerator of the first fraction (3) by the denominator of the second fraction (3). This gives you \(3 \times 3\).
• Multiply the denominator of the first fraction (7) by the numerator of the second fraction \(x\). This gives you \(7 \times x\).
• You now have the equation: \(3 \times 3 = 7 \times x\), or simplified as \(9 = 7x\).

Cross-multiplication simplifies the problem into an equation that is much easier to solve for the unknown value. This technique is fundamental in algebra, especially when solving problems involving ratios and proportions.

Solving Proportions

Solving proportions means finding the value of the variable that makes the two ratios equal. In the equation \(\frac{3}{7} = \frac{x}{3}\), solving the proportion is about finding the value of \(x\). Here's how you can approach it:

• First, use cross-multiplication to transform the proportion into a straightforward equation: \(9 = 7x\).
• Next, isolate the variable \(x\) to solve for it. In this case, divide both sides of the equation by 7 to find \(x\): \( x = \frac{9}{7} \).

Solving proportions requires you to recognize the structure of a proportion and how to manipulate it through cross-multiplication. Implementing these steps helps you find the missing term efficiently.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This means the numerator and the denominator have no common factors other than 1. After determining \(x\) in the equation \(x = \frac{9}{7}\), you assess if it needs further simplification. Here’s how you simplify fractions:

• Check for common factors between the numerator (9) and the denominator (7). Since both numbers are prime relative to each other, the fraction \(\frac{9}{7}\) is already in its lowest terms.
• If there were common factors, you would divide both the numerator and the denominator by the greatest common factor.

Understanding how to simplify fractions ensures your solutions are presented in the most concise form, which is crucial for clarity and conciseness in mathematics.