Problem 69 - Algebra 2

Question

PREREQUISITE SKILL Solve each proportion. $$ \frac{6}{15}=\frac{8}{S} $$

Step-by-Step Solution

Answer

The value of \( S \) is 20.

Step 1: Cross-Multiply

Cross-multiply the two fractions to create an equation. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and setting it equal to the product of the denominator of the first fraction and the numerator of the second fraction.The equation formed will be:\[ 6 imes S = 15 imes 8 \]

Step 2: Solve the Equation

Simplify the equation obtained in Step 1 by performing the multiplication on the right side:\[ 6 imes S = 120 \]Next, solve for \( S \) by dividing both sides of the equation by 6:\[ S = \frac{120}{6} \]

Step 3: Simplify the Solution

Perform the division on the right side to find the value of \( S \):\[ S = 20 \]

Key Concepts

Cross-Multiplication

Cross-multiplication is a fundamental technique used to solve proportions, which are equations that show two ratios are equal. To use cross-multiplication, we follow these steps:

• Take the numerator from the first fraction and multiply it by the denominator of the second fraction.
• Do the same with the numerator of the second fraction and the denominator of the first fraction.
• Set the two products equal to each other to form an equation.

For instance, given the proportion \( \frac{6}{15} = \frac{8}{S} \), we would multiply 6 by \( S \) and 15 by 8, forming the equation \( 6 \times S = 15 \times 8 \). This step is crucial because it helps us eliminate the fractions and work with a simpler equation to find the unknown value.

Solving Equations

Once you have formed an equation using cross-multiplication, the next step is to solve it. Solving equations involves finding the value of an unknown variable that makes the given statement true. Here's how it works in our example:

• First, perform the multiplication indicated in the equation. For \( 6 \times S = 15 \times 8 \), simplify the right-hand side to get \( 120 \).
• The equation then becomes \( 6 \times S = 120 \).
• To isolate \( S \), divide both sides by 6, resulting in \( S = \frac{120}{6} \).

This process of simplifying and rearranging allows us to solve for \( S \), finding the value that will satisfy the original proportion.

Rational Expressions

Rational expressions are essentially fractions that involve variables. In the context of proportions, each side of the equation can be seen as a rational expression. Understanding how to manage these expressions is important for precise solutions. In our example:

• The original proportions \( \frac{6}{15} \) and \( \frac{8}{S} \) are rational expressions because they include numbers and possibly variable denominators or numerators.
• The goal is to manipulate these expressions to solve for the unknown variable, \( S \).
• Cross-multiplication is particularly useful here as it simplifies the problem, converting rational expressions into a solvable linear equation without fractions.

By getting comfortable with rational expressions, you'll find it easier to handle other algebraic problems that involve fractions and unknown variables.