Problem 28 - Prealgebra

Question

Solve each proportion. $$\frac{a}{0.28}=\frac{4}{1.4}$$

Step-by-Step Solution

Answer

The solution to the proportion is \( a = 0.8 \).

Step 1: Find Cross-Products

In a proportion, the cross-products are equal. That means the cross-product of one fraction should equal the cross-product of the other fraction. For the given proportion \( \frac{a}{0.28} = \frac{4}{1.4} \), write the equation: \[ a imes 1.4 = 0.28 imes 4 \]

Step 2: Calculate Cross-Products

Calculate the cross-products: \[ 0.28 imes 4 = 1.12 \] So, the equation becomes: \[ a imes 1.4 = 1.12 \]

Step 3: Solve for Variable

Solve for \( a \) by dividing both sides by 1.4: \[ a = \frac{1.12}{1.4} \] By performing the division, find: \[ a = 0.8 \]

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a key concept when dealing with proportions. This method allows you to solve equations that revolve around fractions. Imagine you are given a proportion like \( \frac{a}{0.28} = \frac{4}{1.4} \). This equation tells us that the ratio of \( a \) to 0.28 is the same as 4 to 1.4. In simple terms, proportions express how two fractions are equivalent.
The cross-multiplication rule states that if two ratios are equal, the cross-products of these ratios are also equal. This means you multiply the numerator of the first fraction by the denominator of the second fraction, and the denominator of the first fraction by the numerator of the second. For our exercise:

• Multiply \( a \times 1.4 \)
• Multiply \( 0.28 \times 4 \)

The cross-products become \( a \times 1.4 = 0.28 \times 4 \). By setting these products equal, you can find the unknown value by solving a simpler equation.

Solving Equations Using Cross-Multiplication

Once you understand cross-multiplication, solving equations involving proportions becomes straightforward. After calculating the cross-products, you'll have an equation that you need to solve for the variable. In our example, you determine that \( a \times 1.4 = 1.12 \). The next step is isolating the variable \( a \). This is accomplished by performing an operation that allows you to get \( a \) alone on one side of the equation. Typically, you do this by using the inverse of the operation affecting \( a \).
In this case:

• Divide both sides by 1.4

This gives us \( a = \frac{1.12}{1.4} \). Performing this division finds that \( a = 0.8 \). This systematic approach ensures you maintain balance in the equation while revealing the unknown value.

Exploring Prealgebra Concepts

Prealgebra forms the foundational skills needed for algebra and beyond. It involves understanding basic math operations and applying them to solve real-world problems, such as proportions. In solving proportions like \( \frac{a}{0.28} = \frac{4}{1.4} \), prealgebra skills enable you to break down the problem into manageable parts.Let's look at the key elements:

• Understanding fractions and their equivalence
• Applying multiplication and division operations
• Using logical reasoning to find unknowns

Prealgebra hinges on grasping these simple concepts and knowing how to apply them correctly to more complex problems. Whether it's simplifying fractions or rearranging an equation, these basic principles guide you through solving equations using methods like cross-multiplication. Ensuring a strong prealgebra foundation will be immensely helpful as you advance in mathematics.