Problem 16 - Prealgebra

Question

Solve each proportion. $$\frac{2}{9.4}=\frac{0.2}{v}$$

Step-by-Step Solution

Answer

The value of \( v \) is 0.94.

Step 1: Set Up the Equation

We have the proportion \( \frac{2}{9.4} = \frac{0.2}{v} \). The goal is to solve for \( v \). Start by setting the two cross products equal to each other, which comes from cross-multiplying: \( 2 \times v = 0.2 \times 9.4 \).

Step 2: Calculate the Right Side

Calculate \( 0.2 \times 9.4 \):\[ 0.2 \times 9.4 = 1.88 \].Substitute this value back into the equation, so you have: \( 2v = 1.88 \).

Step 3: Solve for \( v \)

Now, isolate \( v \) by dividing both sides of the equation by 2:\[ v = \frac{1.88}{2} \].Calculate the result to get \( v = 0.94 \).

Key Concepts

Cross-Multiplication: The Key to Solving Proportions

Cross-multiplication is a powerful technique used to solve proportions, which are equations where two ratios are set equal to each other. In a proportion, the form is \( \frac{a}{b} = \frac{c}{d} \). To solve such an equation for one of the variables, cross-multiplication is the go-to method.

The idea behind cross-multiplication is to eliminate the fractions by multiplying across the diagonal. This means you multiply the numerator of one ratio by the denominator of the other. In the given exercise, we had the proportion \( \frac{2}{9.4} = \frac{0.2}{v} \). By cross-multiplying, we set \( 2 \times v \) equal to \( 0.2 \times 9.4 \).

• First, identify the pairs: \( 2 \) and \( v \); \( 0.2 \) and \( 9.4 \).
• Then, multiply them across: \( 2 \times v \) and \( 0.2 \times 9.4 \).

This mathematical step essentially removes the fractions and simplifies the proportion into a more manageable equation. Cross-multiplication streamlines the process of solving proportions and is particularly helpful when dealing with decimals or complex numbers.

Prealgebra Foundations: Building Blocks for Algebra

Prealgebra serves as the fundamental stepping stone to the world of algebra. It includes basic mathematical skills, concepts, and operations that are necessary for higher-level math. In the exercise we analyzed, solving a proportion is one such prealgebraic concept.

Before diving into algebra equations, it is essential to understand:

• Basic operations: addition, subtraction, multiplication, and division
• Manipulating decimals and fractions
• Recognizing equivalent ratios and understanding proportions

These skills lay the groundwork for more advanced mathematics. By mastering prealgebra, students become equipped to handle algebraic equations with confidence. Concepts like cross-multiplication become easier when the foundational knowledge is solidified.

Understanding and Solving Equations

Equations are mathematical statements that assert the equality of two expressions. The ultimate goal is to find the unknown variable that satisfies the equation. In our exercise, the main aim was to solve for \( v \) in the proportion \( \frac{2}{9.4} = \frac{0.2}{v} \).

After setting up the cross-multiplication, we arrived at the equation \( 2v = 1.88 \). To isolate \( v \), we perform an algebraic operation: division. This is how it works:

• Divide both sides of the equation by the coefficient of \( v \), which is \( 2 \).
• This gives us \( v = \frac{1.88}{2} \).

Carrying out the division, we find that \( v = 0.94 \). Equations allow us to systematically find unknown values, and by using logical steps like division and understanding balance, solving becomes straightforward. Learning to solve equations equips students with problem-solving skills valuable not in just math, but in real-life applications as well.