Problem 17 - 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. $$\frac{3}{7}=\frac{3}{x}$$

Step-by-Step Solution

Answer

The missing term is 7.

Step 1: Set Up the Proportion Equation

The given proportion is \( \frac{3}{7} = \frac{3}{x} \). This is an equation where the two ratios should be equal, implying that their cross-products are equal.

Step 2: Cross-Multiplication

We apply cross multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction: \( 3 \cdot x = 7 \cdot 3 \).

Step 3: Simplify the Equation

The equation from the previous step is \( 3x = 21 \). This is a simple linear equation that we can solve for \( x \).

Step 4: Solve for x

To isolate \( x \), divide both sides of the equation by 3. This gives us \( x = \frac{21}{3} \).

Step 5: Simplify the Fraction

Simplify \( \frac{21}{3} \) by dividing both the numerator and the denominator by the greatest common divisor, which is 3. This results in \( x = 7 \).

Key Concepts

Understanding Cross Multiplication

Cross multiplication is a useful technique for solving proportions. A proportion is an equation where two ratios are equivalent. For example, in the proportion \( \frac{3}{7} = \frac{3}{x} \), cross multiplication helps check and solve for the unknown.
Here's how it works:

• Take the numerator of the first fraction and multiply it by the denominator of the second fraction.
• Then, take the denominator of the first fraction and multiply it by the numerator of the second fraction.
• Set the two products equal to each other.

When cross multiplying, it is essential to ensure that the equation remains balanced. For instance, multiplying across and setting up \(3 \times x = 7 \times 3\) ensures that both sides are equal. This is the foundation needed to continue solving the problem.

Solving Linear Equations

The method of cross multiplication often leads to a linear equation. A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable.
In our case, after cross multiplying, we arrive at the equation \(3x = 21\). This is a simple linear equation where the goal is to solve for \(x\).
Solving involves a few straightforward steps:

• First, isolate the variable by performing the same operation on both sides of the equation.
• In this example, divide both sides by 3 to solve for \(x\).
• This operation simplifies the equation to \(x = \frac{21}{3}\).

The process of isolating the variable is crucial. It enables you to clearly see the value of \(x\) in a step-by-step manner.

Fractions in Lowest Terms

Writing fractions in lowest terms is essential for clarity and simplicity. A fraction is in its lowest terms when the numerator and the denominator have no common factors other than 1.
In the equation \(x = \frac{21}{3}\), both the numerator (21) and the denominator (3) share a common factor of 3. Simplifying involves dividing both by their greatest common divisor (GCD).
Here's how to simplify:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• For this fraction, divide 21 and 3 by 3 to get \(\frac{21}{3} = 7\).

This process of reduction ensures that the fraction is in its simplest form, making the result easier to understand and work with.