Problem 120 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operations. Begin by performing operations in parentheses. $$\left(\frac{1}{2}+\frac{1}{4}\right) \div\left(\frac{1}{2}+\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The result of the operation is \( \frac{9}{10} \)

Step 1: Compute the Addition within Parentheses

Begin by performing the addition within the parentheses: \( \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \) and \( \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \). The operation now looks like this: \( \frac{3}{4} \div \frac{5}{6} \)

Step 2: Change Division to Multiplication

Next, remember that dividing by a fraction is equivalent to multiplying by its reciprocal. Invert the second fraction \( \frac{5}{6} \) to its reciprocal \( \frac{6}{5} \). So, the operation now changes to: \( \frac{3}{4} * \frac{6}{5} \)

Step 3: Multiply the Fractions

Now, multiply the fractions by simply multiplying the numerator with the numerator and the denominator with the denominator: \( \frac{3}{4} * \frac{6}{5} = \frac{18}{20} \)

Step 4: Simplify the Result

Finally, simplify the fraction \( \frac{18}{20} \) to its simplest form, which is \( \frac{9}{10} \)

Key Concepts

Adding Fractions

Adding fractions is like finding a common language so they can work together. When adding fractions, it’s important to have the same denominator, which is known as the common denominator. This means the parts of the fractions need to be divided into equal parts.

For example, when we need to add \( \frac{1}{2} \) and \( \frac{1}{4} \), we start by finding a common denominator. Here, the smallest number that both 2 and 4 can divide into is 4. Notice that the second fraction already has this denominator, so we only need to change the first fraction \( \frac{1}{2} = \frac{2}{4} \). Now that the fractions are \( \frac{2}{4} \) and \( \frac{1}{4} \), they can be added easily:

• Keep the common denominator, which is 4.
• Add the numerators, so \( 2 + 1 = 3 \).

This addition results in \( \frac{3}{4} \).

It’s like speaking the same language—once the denominators match, let the numerators add up!

Dividing Fractions

Dividing fractions involves flipping the second fraction and then multiplying. It might sound strange, but here's why it works. When you divide by a fraction, you're essentially saying, "How many of those fractions fit into the other?" Instead, we make the problem simpler by multiplying.

In our exercise, we have \( \frac{3}{4} \div \frac{5}{6} \). To divide, we'll change this division into multiplication:

• Take \( \frac{5}{6} \), the divisor, and flip it, making it \( \frac{6}{5} \).
• Multiply the result from \( \frac{3}{4} \) by \( \frac{6}{5} \).

After flipping and preparing to multiply, we see it’s simpler to multiply these fractions.

Multiplying Fractions

Multiplying fractions is a straightforward task. The rule is simple: Multiply the numerators together to get a new numerator, and multiply the denominators together to get a new denominator.

So, when multiplying \( \frac{3}{4} \) by \( \frac{6}{5} \):

• Multiply the numerators: \( 3 \times 6 = 18 \).
• Multiply the denominators: \( 4 \times 5 = 20 \).

The result of this operation is \( \frac{18}{20} \), which is our fraction before simplifying.

Remember, there’s no need to worry about finding common denominators here; just multiply straight across!

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and work with.

To simplify: Take the fraction \( \frac{18}{20} \) from our previous multiplication. Both 18 and 20 can be divided by 2, which is their greatest common factor (GCF). Divide both the numerator and the denominator by 2:

• \( 18 \div 2 = 9 \)
• \( 20 \div 2 = 10 \)

Now we have \( \frac{9}{10} \) as our simplified fraction.

Simplifying fractions will leave you with a cleaner, neat answer, simplifying further operations and interpretations.