Problem 73 - Algebra A Combined Function

Question

Perform each indicated operation. $$ \frac{13}{20} \div \frac{2}{9} $$

Step-by-Step Solution

Answer

\( \frac{13}{20} \div \frac{2}{9} = \frac{117}{40} \)

Step 1: Identify the Operation

The operation to be performed is division between two fractions: \( \frac{13}{20} \) and \( \frac{2}{9} \).

Step 2: Find the Reciprocal

To divide by a fraction, we multiply by its reciprocal. The reciprocal of \( \frac{2}{9} \) is \( \frac{9}{2} \).

Step 3: Convert Division to Multiplication

Rewrite the expression by replacing the division with multiplication by the reciprocal: \( \frac{13}{20} \times \frac{9}{2} \).

Step 4: Multiply the Fractions

Multiply the numerators together and the denominators together: \( \frac{13 \times 9}{20 \times 2} = \frac{117}{40} \).

Step 5: Simplify the Result

The fraction \( \frac{117}{40} \) is in its simplest form since 117 and 40 have no common factors apart from 1.

Key Concepts

Understanding Reciprocals

When working with fractions, especially in division, the idea of a **reciprocal** is incredibly important. The reciprocal of a number is what you multiply it by to get 1, also known as the multiplicative inverse.

For instance, if you start with a fraction like \( \frac{2}{9} \), its reciprocal would be simply flipping the numerator and the denominator, giving you \( \frac{9}{2} \). This is because when you multiply \( \frac{2}{9} \) and \( \frac{9}{2} \), you get 1. Here's how that looks:

• \( \frac{2}{9} \times \frac{9}{2} = \frac{2 \times 9}{9 \times 2} = \frac{18}{18} = 1 \)

Reciprocals are key because they allow us to turn division into multiplication, which is typically easier to manage.

Multiplying Fractions Made Easy

Multiplying fractions is a straightforward process once you understand the steps. To multiply fractions, follow these steps:

• Multiply the numerators (the top numbers) together.
• Multiply the denominators (the bottom numbers) together.

For example, when you have \( \frac{13}{20} \times \frac{9}{2} \), you perform the multiplication like this:

• The numerators: \( 13 \times 9 = 117 \)
• The denominators: \( 20 \times 2 = 40 \)

This results in the fraction \( \frac{117}{40} \). Just like that, multiplication of fractions is complete, and now you have a new fraction!

Simplifying Fractions

After multiplying fractions, we often want to simplify the result. Simplifying makes the fraction easier to understand and work with. A fraction is simplified when the numerator and denominator do not share any common factors other than 1.

In our example with \( \frac{117}{40} \), we look to see if these two numbers have any shared factors.

• First, identify any common factors. Here, 117 and 40 do not share factors, meaning the simplest form is \( \frac{117}{40} \).

It’s essential to remember that if common factors exist, you divide the numerator and the denominator by the largest common factor to simplify. Simplifying fractions may not always change the fraction's value, but it can make calculations easier in the long run.