Problem 34 - Prealgebra

Question

Suppose \(a: b=2: 4\) and \(b: c=4: 9 .\) If \(a=10,\) find the value of \(c .\)

Step-by-Step Solution

Answer

The value of \(c\) is 45.

Step 1: Understand the Problem

We are given ratios \(a:b = 2:4\) and \(b:c = 4:9\) and we know that \(a = 10\). We need to find the value of \(c\).

Step 2: Express Ratios as Fractions

First, express the ratios as fractions. From \(a:b = 2:4\), we can say \(\frac{a}{b} = \frac{2}{4}\). Similarly, from \(b:c = 4:9\), we have \(\frac{b}{c} = \frac{4}{9}\).

Step 3: Solve for b using a:b Equation

Since \(\frac{a}{b} = \frac{2}{4}\), we substitute \(a = 10\) to get \(\frac{10}{b} = \frac{2}{4}\). Cross-multiply to solve for \(b\):\[ 2b = 10 \times 4 \]\[ 2b = 40 \]\[ b = 20 \]

Step 4: Solve for c using b:c Equation

Now that we have \(b = 20\), use the equation \(\frac{b}{c} = \frac{4}{9}\). Substitute \(b = 20\) to get:\[ \frac{20}{c} = \frac{4}{9} \]Cross-multiply to find \(c\):\[ 4c = 20 \times 9 \]\[ 4c = 180 \]\[ c = \frac{180}{4} \]\[ c = 45 \]

Step 5: Summarize the Result

We have found that when \(a = 10\), the corresponding value of \(c\) is 45.

Key Concepts

Fraction Operations

Fractions are a way of expressing numbers that are not whole. They consist of a numerator and a denominator. In any fraction operation, stick to the basic rules: add or subtract fractions with a common denominator, and multiply or divide by using cross-multiplication.

In the context of ratios, fractions help to express and manipulate these relationships. For instance, when given a ratio like \(a:b = 2:4\), this means for every 2 units of \(a\), there are 4 units of \(b\). We represent it as a fraction \(\frac{a}{b} = \frac{2}{4}\).

Simplifying or manipulating these fractions allows us to solve more complex problems by providing a clear comparison metric.

Cross-Multiplication

Cross-multiplication is a powerful tool when dealing with equations involving fractions. This method involves multiplying diagonally across the equals sign.

For example, with the equation \(\frac{a}{b} = \frac{2}{4}\) and \(a = 10\), we cross-multiply to solve for \(b\). We multiply 2 and \(b\) and 4 and 10, yielding the equation \(2b = 40\), which we then solve for \(b\).

Similarly, when solving \(\frac{b}{c} = \frac{4}{9}\), cross-multiplying (\(4c = 20 \times 9\)) helps us isolate and solve for \(c\). This technique is useful because it eliminates fractions from the equation and transforms it into a simple multiplication problem.

Prealgebra Problem-Solving

Prealgebra serves as the foundation for all higher levels of algebra and is essential for solving real-world problems. It involves understanding basic mathematical concepts and applying them to everyday scenarios. It emphasizes the ability to think logically and solve equations efficiently.

The given problem of finding \(c\) involves understanding ratios and proportions, converting them into fractions, and then using algebraic techniques such as cross-multiplication for the solution.

The steps to solve these problems are generally as follows:

• Interpret the problem to understand what is being asked.
• Translate the given ratios or relationships into fractions or equations.
• Choose the correct mathematical operations or techniques to simplify and solve these equations.
• Verify your solution by substituting back into the original context to ensure consistency.

This process builds problem-solving skills and prepares students for more advanced mathematical challenges.