Problem 44 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{3}{4} \div 6$$

Step-by-Step Solution

Answer

\(\frac{1}{8}\)

Step 1: Understand Division of Fractions

When dividing fractions, we multiply by the reciprocal of the divisor. In this problem, we have \(\frac{3}{4} \div 6\). We first need to express \(6\) as a fraction to perform the division.

Step 2: Express Divisor as a Fraction

The number \(6\) can be written as \(\frac{6}{1}\). This helps us to perform the division using the multiplication by reciprocal method.

Step 3: Find the Reciprocal of the Divisor

The reciprocal of \(\frac{6}{1}\) is \(\frac{1}{6}\). We will use this reciprocal to convert the division into a multiplication problem.

Step 4: Perform the Multiplication

Now, multiply \(\frac{3}{4}\) by the reciprocal \(\frac{1}{6}\): \[ \frac{3}{4} \times \frac{1}{6} = \frac{3 \times 1}{4 \times 6} = \frac{3}{24} \].

Step 5: Simplify the Fraction

The resulting fraction is \(\frac{3}{24}\). Simplify by dividing the numerator and the denominator by their greatest common divisor, which is 3: \[ \frac{3 \div 3}{24 \div 3} = \frac{1}{8} \].

Key Concepts

Reciprocal

The concept of a reciprocal is foundational to fraction division. A reciprocal of a number is what you multiply the number by to get 1. For any non-zero number or fraction, its reciprocal is calculated by flipping the numerator with the denominator.
For instance, in the fraction \( \frac{6}{1} \), which is the fractional representation of 6, the reciprocal is \( \frac{1}{6} \).

• To find the reciprocal of a whole number, represent it as a fraction first. This involves putting 1 as the denominator.
• Then, swap the numerator and denominator. This mathematically represents flipping the fraction.

Using reciprocals turns a division problem into a multiplication one, making calculations straightforward, especially with fractions.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. This means making the numerator and denominator as small as possible while retaining the value of the fraction.
For the example \( \frac{3}{24} \), both 3 and 24 can be divided by their greatest common divisor (GCD), which is 3.

• First, identify the GCD of the numerator and the denominator.
• Then, divide both the numerator and the denominator by the GCD.
• Here, dividing by 3 gives \( \frac{3 \div 3}{24 \div 3} = \frac{1}{8} \).

This process simplifies fractions, making them easier to understand and use. Sometimes, the fraction might already be in its simplest form, in which case no changes are needed.

Division of Whole Numbers by Fractions

When you divide whole numbers by fractions, it involves a few simple yet crucial steps.
First, express the whole number as a fraction by giving it a denominator of 1. For example, 6 becomes \( \frac{6}{1} \).
Then, apply the concept of reciprocals.

• Find the reciprocal of the fraction you are dividing by.
• In our exercise, for \( \frac{6}{1} \), the reciprocal becomes \( \frac{1}{6} \).
• Next, convert the division into multiplication by multiplying the original fraction by this reciprocal.
• Multiply \( \frac{3}{4} \) by \( \frac{1}{6} \) to get \( \frac{3}{24} \).

Finally, simplify the resulting fraction to get to the answer, \( \frac{1}{8} \).
This straightforward approach of using reciprocals makes dividing with fractions easier to handle.