Problem 59 - Prealgebra

Question

Find quotient. Write in simplest form. \(\frac{1}{6} \div \frac{3}{4}\)

Step-by-Step Solution

Answer

The quotient in simplest form is \(\frac{2}{9}\).

Step 1: Understanding the Division of Fractions

To divide fractions, you need to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.

Step 2: Finding the Reciprocal

The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). Swap the numerator (3) and the denominator (4) to get the reciprocal.

Step 3: Multiplying the Fractions

Now multiply the first fraction by the reciprocal of the second: \(\frac{1}{6} \times \frac{4}{3}\). Multiply the numerators and the denominators separately: \(1 \times 4 = 4\) and \(6 \times 3 = 18\). This gives \(\frac{4}{18}\).

Step 4: Simplifying the Fraction

To simplify \(\frac{4}{18}\), find the greatest common divisor (GCD) of 4 and 18, which is 2. Divide both the numerator and the denominator by 2: \(\frac{4 \div 2}{18 \div 2} = \frac{2}{9}\).

Step 5: Final Result

The simplest form of the quotient is \(\frac{2}{9}\).

Key Concepts

Division of Fractions

Dividing fractions might seem tricky at first, but it's practically a two-step process. The key to dividing fractions is to "flip and multiply." The term "flip" refers to finding the reciprocal of the second fraction. Once you've got that, you multiply it with the first fraction.

• First, understand the problem: Given the example \(\frac{1}{6} \div \frac{3}{4}\), your task is to divide the fractions.
• Remember the rule: To divide by a fraction, multiply by its reciprocal.
• The reciprocal of a fraction is simply the original fraction flipped upside down. More about that in the following sections.
• After finding the reciprocal, multiply the original fraction (\(\frac{1}{6}\)) with this new reciprocal (\(\frac{4}{3}\)).

The outcome of this multiplication gives you the answer to the division problem. Once it's clear how to divide fractions, you'll find it as straightforward as multiplying whole numbers.

Simplifying Fractions

Simplifying fractions is about making them as simple as possible without changing their value. Once you've performed the division of fractions, your result might not always be in its simplest form. Certain steps can help streamline this process:

• After the division problem, you might end up with something like \(\frac{4}{18}\).
• To simplify, you need to find a number that both the numerator (4) and the denominator (18) can be divided by evenly. This number is known as the greatest common divisor (GCD).
• For 4 and 18, the GCD is 2. Divide both the numerator and the denominator by their GCD.
• \(\frac{4}{18} \div \frac{2}{2} = \frac{2}{9}\).

Now, \(\frac{2}{9}\) is in its simplest form. Its value has not changed, just its presentation. Simplification makes fractions more comprehensible and easier to work with.

Reciprocal of a Fraction

The reciprocal of a fraction is a concept that flips the fraction's structure. This means exchanging the places of the numerator and the denominator. Recognizing and finding the reciprocal is crucial in operations like division.

• The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).
• In our example, the fraction \(\frac{3}{4}\) becomes its reciprocal \(\frac{4}{3}\).
• Swapping these two parts is essential when dividing fractions because the operation requires multiplication by the reciprocal, making the division process possible.
• This nifty trick simplifies division considerably, making hard problems much easier to solve.

Learning how to quickly find the reciprocal is a stepping stone for efficient fraction division. Once you master this, fraction operations become a breeze!