Problem 171 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{\frac{m}{3}}{\frac{n}{2}} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{2m}{3n}\).

Step 1 - Write the Original Expression

The given expression is \[\frac{\frac{m}{3}}{\frac{n}{2}}\].

Step 2 - Apply the Division Rule for Fractions

When dividing two fractions, such as \(\frac{a}{b} \div \frac{c}{d}\), the result is equivalent to multiplying by the reciprocal of the second fraction: \(\frac{a}{b} \times \frac{d}{c}\). Therefore, \[\frac{\frac{m}{3}}{\frac{n}{2}} = \frac{m}{3} \times \frac{2}{n}\].

Step 3 - Multiply the Fractions

Multiply the numerators and the denominators: \[\frac{m \times 2}{3 \times n} = \frac{2m}{3n}\].

Step 4 - Simplify the Result

The expression is already simplified. The simplified form is \[\frac{2m}{3n}\].

Key Concepts

Division of Fractions

Dividing fractions might seem tricky at first, but it's simpler than you think. The key is to understand that dividing by a fraction is the same as multiplying by its reciprocal. Let's break down these concepts to make things clear.
When you have an expression like \(\frac{\frac{m}{3}}{\frac{n}{2}}\), you need to change the division into multiplication by using the reciprocal of the second fraction. The reciprocal of a fraction \( \frac{a}{b} \) is simply \( \frac{b}{a} \). So, \(\frac{n}{2}\) becomes \( \frac{2}{n} \) when flipped.
With this understanding, we change the problem into a multiplication one: \(\frac{m}{3} \times \frac{2}{n}\). Now you're ready to dive into the multiplication of fractions!

Multiplication of Fractions

Multiplying fractions is straightforward once you know the reciprocal concept. To multiply two fractions, you simply multiply their numerators and then their denominators.
For example, in our expression \(\frac{m}{3} \times \frac{2}{n}\), multiply the numerators (the top numbers) and then the denominators (the bottom numbers):

• Multiply the numerators: \m \times 2 = 2m\
• Multiply the denominators: \3 \times n = 3n\

The multiplication gives you \(\frac{2m}{3n}\).
It's that simple! Just make sure to always multiply straight across when dealing with fractions.

Reciprocal

Understanding the reciprocal is crucial in fraction division. A reciprocal of a number is simply 1 divided by that number. For any fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
The reciprocal changes when you flip the numerator and the denominator. This flipping is what transforms the division problem into a multiplication one.
For instance, in \(\frac{\frac{m}{3}}{\frac{n}{2}}\), the reciprocal of \(\frac{n}{2}\) is \(\frac{2}{n}\). By converting the problem into multiplication through reciprocals, we simplify our work and better understand the operation.
So, always remember: finding the reciprocal is just flipping the fraction!