Problem 70 - Algebra 2

Question

PREREQUISITE SKILL Solve each proportion. $$ \frac{b}{9}=\frac{40}{30} $$

Step-by-Step Solution

Answer

The solution is \(b = 12\).

Step 1: Cross-Multiply

Start by cross-multiplying the two fractions to eliminate the denominators. Multiply the numerator of the first fraction by the denominator of the second fraction, and similarly multiply the denominator of the first fraction by the numerator of the second fraction.This gives: \[ b \times 30 = 9 \times 40 \]

Step 2: Simplify the Equation

Now, perform the multiplication on each side of the equation. Calculate \(9 \times 40\). This simplifies to: \[ 30b = 360 \]

Step 3: Solve for b

To isolate \(b\), divide both sides of the equation by 30:\[ b = \frac{360}{30} \]

Step 4: Simplify the Result

Perform the division to find the value of \(b\). Solve \(\frac{360}{30}\) by dividing 360 by 30:This equals:\[ b = 12 \]

Key Concepts

Cross Multiplication

Cross multiplication is a handy tool used to solve proportions, which are equations that show two fractions are equal. When you have a proportion like \( \frac{b}{9} = \frac{40}{30} \), you can cross-multiply to eliminate the fractions. To cross-multiply, you multiply the numerator of one fraction by the denominator of the other fraction. Then, do the same with the other numerator and denominator.
For our example, we multiply \( b \) by 30 and 9 by 40, leading to the equation:

• \( b \times 30 = 9 \times 40 \)

This step is critical because it transforms the proportion into a simple equation without fractions, making it easier to solve. Cross-multiplication relies on the property that if two fractions are equal, then their cross products are also equal. This is why it's such a useful technique for solving proportions.

Solving Equations

Once cross-multiplication has been applied to get \( 30b = 360 \), the task becomes solving the resulting equation. Solving equations involves finding the value of the variable (in this case, \( b \)) that makes the equation true.
The equation \( 30b = 360 \) is a simple linear equation where \( b \) is multiplied by 30. To isolate \( b \), you need to perform the opposite operation to multiplication, which is division.

• Divide both sides of the equation by 30: \( \frac{30b}{30} = \frac{360}{30} \)

When you divide both sides by 30, you simplify to get \( b = 12 \). This process of solving involves applying basic operation reversals to isolate the variable and determine its value. Remember, whatever operation you do to one side of the equation, you must do to the other to keep it balanced.

Simplifying Fractions

Simplifying fractions is an essential skill that makes numbers easier to handle and understand. Once \( b = \frac{360}{30} \) was obtained, simplifying the fraction shows you the most straightforward form of \( b \).
Simplifying means reducing the fraction by dividing both the top (numerator) and bottom (denominator) by their greatest common divisor (GCD). For \( \frac{360}{30} \):

• The GCD of 360 and 30 is 30.
• Divide the numerator and denominator by 30.
• \( \frac{360 \div 30}{30 \div 30} = \frac{12}{1} = 12 \)

Simplifying turns the fraction into a whole number, making it clearer and more comprehensible. In practice, simplifying fractions helps streamline calculations and reduces potential errors in problem-solving.