Problem 110 - Introductory and Intermediate Algebra for College Students 4th

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 result of the division \( \frac{2}{3} \div \frac{7}{5} \) is \( \frac{10}{21} \)

Step 1: Identify the Given Fractions

The given fractions are \( \frac{2}{3} \) and \( \frac{7}{5} \). The former is to be divided by the latter.

Step 2: Write the Operation as Multiplication

As per the rule of division for fractions, this operation can be transformed into a multiplication operation by taking the reciprocal of the second fraction. Thus, we treat the operation \( \frac{2}{3} \div \frac{7}{5} \) as \( \frac{2}{3} \cdot \frac{5}{7} \).

Step 3: Perform the Multiplication

Now, we proceed with the multiplication as usual for fractions: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. Thus, \( \frac{2}{3} \cdot \frac{5}{7} = \frac{2 \cdot 5}{3 \cdot 7} = \frac{10}{21} \).

Key Concepts

Fraction Division

Understanding how to divide fractions is a crucial skill in mathematics. Let's say you're given two fractions that you need to divide, such as \( \frac{2}{3} \) divided by \( \frac{7}{5} \). The process might seem complicated at first, but it's actually quite simple once you know the right steps.

In dividing fractions, you'll want to remember a core principle: division of fractions is equivalent to multiplying by the reciprocal. This means that to divide one fraction by another, you actually multiply the first fraction by the reciprocal of the second.

For example, to divide \( \frac{2}{3} \) by \( \frac{7}{5} \), you multiply \( \frac{2}{3} \) by the reciprocal of \( \frac{7}{5} \), which is \( \frac{5}{7} \). This turns a division problem into a multiplication problem, making it easier to solve.

Reciprocal of a Fraction

The reciprocal of a fraction is simply a way to flip a fraction upside down. In other words, you switch the numerator and the denominator. For instance, the reciprocal of \( \frac{7}{5} \) is \( \frac{5}{7} \).

Finding the reciprocal is a critical step in fraction division because it transforms the operation from division to multiplication. This switch is what makes dividing fractions possible.

Why Reciprocals Matter

Understanding reciprocals is not only essential in fraction division but also in various algebraic processes where you may need to invert coefficients or solve equations involving fractions. When dealing with division, always remember: turn the divisor (the fraction you're dividing by) into its reciprocal and multiply.

Multiplying Fractions

After you've turned the division problem into multiplication by using the reciprocal of the second fraction, multiplying fractions is straightforward. To multiply fractions, multiply the numerators (the top numbers) to find the new numerator and multiply the denominators (the bottom numbers) to find the new denominator.

In our example, to multiply \( \frac{2}{3} \) and \( \frac{5}{7} \), you calculate \(2 \times 5 = 10\) for the numerator and \(3 \times 7 = 21\) for the denominator. This gives us \( \frac{10}{21} \), the product of our two fractions.

Here's a simple tip: Before multiplying, check if you can simplify the fractions by cancelling out any common factors between the numerators and denominators to make the multiplication even easier.