Problem 45 - Prealgebra

Question

Find each quotient. $$\frac{3 y}{4} \div \frac{5 y}{8}$$

Step-by-Step Solution

Answer

\(\frac{3y}{4} \div \frac{5y}{8} = \frac{6}{5}\)

Step 1: Understand the Problem

We are given the division of two fractions: \(\frac{3y}{4}\) and \(\frac{5y}{8}\). We need to find the quotient of these two fractions.

Step 2: Rewrite Division as Multiplication

To divide by a fraction, you can multiply by its reciprocal. So, \(\frac{3y}{4} \div \frac{5y}{8}\) becomes \(\frac{3y}{4} \times \frac{8}{5y}\).

Step 3: Multiply the Fractions

Multiply the numerators together and the denominators together: \[\frac{3y \times 8}{4 \times 5y} = \frac{24y}{20y}\].

Step 4: Simplify the Fraction

Cancel out the common terms in the numerator and denominator. Here, \(y\) can be canceled out because it's present in both, resulting in: \[\frac{24}{20}\].

Step 5: Simplify Further by Reducing the Fraction to Lowest Terms

Divide the numerator and the denominator by their greatest common divisor, which is 4: \[\frac{24 \div 4}{20 \div 4} = \frac{6}{5}\].

Key Concepts

Understanding Reciprocals

The concept of reciprocals is crucial when dealing with fraction division. A reciprocal is a flipped version of the original fraction. For instance, the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).

When you divide by a fraction, like in the given problem \(\frac{3y}{4} \div \frac{5y}{8}\), you multiply by the reciprocal instead. This means you take \(\frac{5y}{8}\) and flip it to become \(\frac{8}{5y}\).

Knowing how to find and use reciprocals makes fraction division straightforward.

Multiplication of Fractions Explained

Once you convert division into multiplication using the reciprocal, the next step is to multiply the fractions. The process of multiplying fractions is simpler than it seems. Here’s how it works in our exercise:

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

In the problem, you have \(\frac{3y}{4} \times \frac{8}{5y}\). So, it becomes \(\frac{3y \times 8}{4 \times 5y} = \frac{24y}{20y}\). Each step follows straightforward multiplication, ensuring you keep fractions manageable.

Simplifying Fractions

Simplifying fractions is about making them as simple as possible without changing their value.

After multiplying, you may end up with a larger fraction. Here, \(\frac{24y}{20y}\), appears right after multiplication. To simplify, you can cancel out any common factors present in both the numerator and the denominator.

In our exercise, \(y\) is common in both and can be canceled leaving you with \(\frac{24}{20}\). Simplifying aids in understanding and ensures fractions presented are in their simplest form.

Greatest Common Divisor (GCD) Use

To fully simplify fractions, using the greatest common divisor (GCD) is indispensable. GCD helps to reduce a fraction to its lowest terms by dividing both the numerator and the denominator by it.

In the final part of our example, \(\frac{24}{20}\) can be reduced. You find the GCD of 24 and 20, which is 4. Dividing both numbers gives \(\frac{24 \div 4}{20 \div 4} = \frac{6}{5}\).

This final result represents the simplest form of the original fraction, making calculations and understanding easier.