Problem 95 - Elementary Algebra

Question

\text { Describe how to divide two fractions. }

Step-by-Step Solution

Answer

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.

Step 1: Understand the Problem

To divide two fractions, keep in mind that dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Identify the Fractions

Identify the two fractions that need to be divided. For example, consider \(\frac{a}{b}\) and \(\frac{c}{d}\).

Step 3: Find the Reciprocal of the Second Fraction

Take the second fraction \(\frac{c}{d}\) and find its reciprocal, which is \(\frac{d}{c}\).

Step 4: Multiply by the Reciprocal

Change the division problem into a multiplication problem by multiplying the first fraction by the reciprocal of the second. This gives \(\frac{a}{b} \times \frac{d}{c}\).

Step 5: Multiply the Numerators

Multiply the numerators of the two fractions. The result is \(a \times d\).

Step 6: Multiply the Denominators

Multiply the denominators of the two fractions. The result is \(b \times c\).

Step 7: Simplify the Resulting Fraction

Simplify the fraction \(\frac{a \times d}{b \times c}\) if possible.

Key Concepts

Reciprocal of a Fraction

A reciprocal of a fraction is an important concept when dividing fractions. The reciprocal simply means flipping the fraction. For instance, if you have the fraction \(\frac{c}{d}\), the reciprocal would be \(\frac{d}{c}\).
The idea behind reciprocals is to turn a division problem into a multiplication problem, making it easier to solve.
When you take the reciprocal, you switch the numerator (top number) and the denominator (bottom number).

• An example: The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
• If you have a whole number like 5, its reciprocal is \(\frac{1}{5}\).

It's essential to be comfortable finding reciprocals because it's a key step in dividing fractions.

Multiplying Fractions

Multiplying fractions might sound complex, but it’s straightforward. Simply multiply the numerators together and the denominators together.
For example, if you’re multiplying \(\frac{a}{b}\) and \(\frac{d}{c}\), you do the following steps:

• Step 1: Multiply the numerators (top numbers): \(a \times d\).
• Step 2: Multiply the denominators (bottom numbers): \(b \times c\).

So the result will be \(\frac{a \times d}{b \times c}\).
This step is useful even when dividing fractions. Remember, dividing fractions involves multiplying by the reciprocal of the second fraction.
For instance, if you have \(\frac{2}{3}\) divided by \(\frac{4}{5}\), you multiply by the reciprocal of \(\frac{4}{5}\), which is \(\frac{5}{4}\). So, \(\frac{2}{3} \times \frac{5}{4}\ = \frac{10}{12}\).
Multiplying fractions is an essential skill, especially when simplifying and solving equations.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This process helps make the fraction easier to work with and understand.
A fraction is simplified when the numerator and denominator have no common factors other than 1. To simplify a fraction, follow these steps:

• Step 1: Find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both evenly.
• Step 2: Divide both the numerator and the denominator by their GCD.

For example, let's simplify \(\frac{10}{12}\):

• The GCD of 10 and 12 is 2.
• So, \(\frac{10}{12}\) becomes \(\frac{10 \div 2}{12 \div 2} = \frac{5}{6}\).

Simplifying fractions helps make calculations easier and comparisons simpler. It’s very useful in solving fraction problems effectively. Remember always to simplify your final answer!