Problem 110 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. In each exercise, use the given formula to perform the indicated operation with the two fractions. $$\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \cdot \frac{d}{c}=\frac{a \cdot d}{b \cdot c} ; \quad \frac{2}{3} \div \frac{7}{5}$$

Step-by-Step Solution

Answer

The division of the fractions \( \frac{2}{3} \) by \( \frac{7}{5} \) is \( \frac{10}{21} \)

Step 1: Understand the Problem

In this problem, we need to evaluate the expression \( \frac{2}{3} \) divided by \( \frac{7}{5} \). We use the formula \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} \) where our \( a \), \( b \), \( c \), and \( d \) are 2, 3, 7, and 5 respectively.

Step 2: Substitute into the Formula

Substitute the values of 2 for \( a \), 3 for \( b \), 7 for \( c \), and 5 for \( d \) in the formula \( \frac{a}{b} \div \frac{c}{d} \) to get \( \frac{2}{3} \div \frac{7}{5} = \frac{2}{3} \cdot \frac{5}{7} \)

Step 3: Perform the Multiplication

Now, we can simply perform the multiplication \( \frac{2}{3} \cdot \frac{5}{7} \). To do this, we multiply the numerators together and the denominators together which gives us \( \frac{2 \cdot 5}{3 \cdot 7} = \frac{10}{21} \)

Key Concepts

Fractions Multiplication

Multiplying fractions might seem challenging at first, but it's actually quite straightforward. When you multiply two fractions, you're essentially multiplying the two numerators and the two denominators.
Here's how it works:

• The top numbers (numerators) are multiplied with each other.
• The bottom numbers (denominators) are also multiplied with each other.

For example, when solving \( \frac{2}{3} \cdot \frac{5}{7} \), you multiply 2 by 5 to get 10 (the new numerator), and 3 by 7 to get 21 (the new denominator). This results in \( \frac{10}{21} \).
Remember, once you've multiplied the fractions, always check if you can simplify them. In our case, \( \frac{10}{21} \) is already in its simplest form.

Rational Expressions

Rational expressions are like fractions, but they contain polynomials in the numerator, denominator, or both. For example, \( \frac{x^2 +1}{x+1} \) is a rational expression.
When working with rational expressions, you can perform arithmetic operations such as addition, subtraction, multiplication, and division just like with numerical fractions.

• To multiply rational expressions, follow the same steps as multiplying fractions: multiply the numerators and multiply the denominators.
• When dividing, invert (flip) the second fraction and multiply.

Working with rational expressions also requires simplifying them. This might involve factoring polynomials and canceling out common factors. Simplification helps make calculations more manageable and solutions clearer. Also, remember that rational expressions are undefined when the denominator is zero, so always check for values that might make the denominator zero.

Step-by-Step Solutions

Step-by-step solutions are a great way to break down complex problems into manageable parts. They help in understanding the procedure needed to arrive at the correct answer and demonstrate the logical flow of the solution.
Here’s why they are important:

• They guide you through each stage of the problem-solving process, ensuring you don’t miss critical steps.
• Step-by-step solutions show how the mathematical rules and formulas are applied in practical scenarios.

Let's take the example of dividing \( \frac{2}{3} \div \frac{7}{5} \). Following the step-by-step approach:
1. **Convert the Division to Multiplication**: Change \( \frac{2}{3} \div \frac{7}{5} \) to \( \frac{2}{3} \cdot \frac{5}{7} \) (by flipping the second fraction and changing the symbol to multiplication).
2. **Multiply**: Use the rules for multiplying fractions to find the product \( \frac{10}{21} \).
This logical progression from start to finish enhances comprehension, especially as you tackle more complex math topics.