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Chapter 1: Problem 391

Simplify. $$ \frac{\frac{m}{3}}{\frac{n}{2}} $$

Short Answer

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

Step by step solution

01

Understand the Problem

The expression is a fraction with fractions in both the numerator and the denominator. We need to simplify this expression.
02

Rewrite the Complex Fraction

Rewrite the complex fraction \(\frac{\frac{m}{3}}{\frac{n}{2}}\) as a division of two fractions: \(\frac{m}{3} \div \frac{n}{2}\).
03

Convert Division to Multiplication

Remember that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \(\frac{m}{3} \div \frac{n}{2}\) can be rewritten as \(\frac{m}{3} \times \frac{2}{n}\).
04

Multiply Fractions

Multiply the numerators together and the denominators together: \(\frac{m \times 2}{3 \times n} = \frac{2m}{3n}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

division of fractions
When you see an expression that involves the division of two fractions, the process can be simplified by turning the division problem into a multiplication problem. This simplifies calculations and makes the steps clearer.

For example, let鈥檚 take the complex fraction: \(\frac{\frac{m}{3}}{\frac{n}{2}}\).

First, rewrite this complex fraction as a division: \(\frac{m}{3} \div \frac{n}{2}\).

This brings us to our next key concept, the reciprocal.
reciprocal
The reciprocal of a fraction is simply what you get when you flip the numerator and the denominator. So, if you have \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).

Why is this important? When dividing fractions, you multiply by the reciprocal of the divisor.

In our exercise, \(\frac{n}{2}\) is the divisor. The reciprocal of \(\frac{n}{2}\) is \(\frac{2}{n}\).

So, \(\frac{m}{3} \div \frac{n}{2}\) becomes \(\frac{m}{3} \times \frac{2}{n}\). This makes the problem more manageable and leads us to the next fundamental step, which is multiplying fractions.
multiplying fractions
Multiplying fractions is straightforward. You multiply the numerators together and the denominators together.

In our exercise, we need to multiply \(\frac{m}{3}\) by \(\frac{2}{n}\).

Numerators: \m \times 2\
Denominators: \3 \times n\

Combine them to get \(\frac{m \times 2}{3 \times n} = \frac{2m}{3n}\).

This simplifies our complex fraction to \(\frac{2m}{3n}\), making the original problem much easier to handle. Remember, always follow these steps: Rewrite, Reciprocal, and Multiply. Happy simplifying!

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