Problem 20 - Prealgebra

Question

Find the following quotients. $$8 \frac{2}{3} \div 4 \frac{1}{3}$$

Step-by-Step Solution

Answer

The quotient is 2.

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. For \(8 \frac{2}{3}\), multiply the whole number 8 by the denominator 3 and add the numerator 2: \(8 \times 3 + 2 = 24 + 2 = 26\). So, \(8 \frac{2}{3}\) becomes \(\frac{26}{3}\). For \(4 \frac{1}{3}\), multiply the whole number 4 by the denominator 3 and add the numerator 1: \(4 \times 3 + 1 = 12 + 1 = 13\). So, \(4 \frac{1}{3}\) becomes \(\frac{13}{3}\).

Step 2: Rewrite the Division as Multiplication

Rewrite the division problem \(\frac{26}{3} \div \frac{13}{3}\) as a multiplication problem by multiplying \(\frac{26}{3}\) by the reciprocal of \(\frac{13}{3}\). The reciprocal of \(\frac{13}{3}\) is \(\frac{3}{13}\). So we have: \(\frac{26}{3} \times \frac{3}{13}\).

Step 3: Multiply the Fractions

Multiply the two fractions together: \(\frac{26}{3} \times \frac{3}{13} = \frac{26 \times 3}{3 \times 13}\). Simplify the multiplication: \(\frac{78}{39}\).

Step 4: Simplify the Resulting Fraction

Simplify the fraction \(\frac{78}{39}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 39. So, \(\frac{78}{39} = \frac{78 \div 39}{39 \div 39} = \frac{2}{1}\), which simplifies further to 2.

Key Concepts

Mixed Numbers

Mixed numbers are numbers that have both a whole number and a fractional part. They appear commonly in real life, such as measuring heights or distances. In mathematical problems, mixed numbers can make calculations a bit trickier than dealing with whole numbers or simple fractions alone.
For example, in your exercise, we had \(8 \frac{2}{3}\) and \(4 \frac{1}{3}\). These are mixed numbers because they include both a whole number and a fraction.
To work with mixed numbers effectively, it's often easier to first convert them into improper fractions. This conversion simplifies arithmetic operations like multiplication or division. To convert a mixed number to an improper fraction:

• Multiply the whole number by the fraction's denominator.
• Add the fraction's numerator to the product.
• Place this sum over the original denominator.

For \(8 \frac{2}{3}\), you would calculate: \(8 \times 3 + 2\), giving you \(\frac{26}{3}\). This transformation makes further calculations simpler.

Improper Fractions

An improper fraction is a type of fraction where the numerator is larger than or equal to the denominator. This happens when the fraction represents a value greater than or equal to one. In the context of solving math problems, improper fractions are often more convenient because they are straightforward to manipulate in arithmetic operations.
When you converted the mixed numbers \(8 \frac{2}{3}\) and \(4 \frac{1}{3}\) into improper fractions, you got \(\frac{26}{3}\) and \(\frac{13}{3}\), respectively. With improper fractions:

• Division can be turned into multiplication.
• It’s easier to compare values at a glance.

Improper fractions allow one to perform further operations such as finding common denominators or simplifying solutions without the hassle of dealing with separate whole and fractional parts.

Reciprocals

The concept of reciprocals is vital in dividing fractions. A reciprocal of a fraction is simply a fraction flipped upside down, swapping the denominator and numerator. In other words, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
Reciprocals play a crucial role in division. Rather than dividing fractions directly, you multiply by the reciprocal of the divisor. This step simplifies the process greatly, turning division problems into multiplication ones, which are typically easier to handle.
In your example, to divide \(\frac{26}{3} \div \frac{13}{3}\), we used the reciprocal of \(\frac{13}{3}\), which is \(\frac{3}{13}\). This allowed the division to transform into multiplication: \(\frac{26}{3} \times \frac{3}{13}\). This procedure helps maintain a steady workflow when dealing with fraction division sections.

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form, where the numerator and denominator are the smallest possible whole numbers that can represent the fraction's value. Simplifying fractions is essential because it makes fractions easier to understand and work with.
To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator. Once identified, divide both by this number.
In your problem, after multiplying the fractions \(\frac{26}{3} \times \frac{3}{13}\) resulting in \(\frac{78}{39}\), simplification was done by identifying the GCD of 78 and 39, which is 39.

• You divide the numerator (78) by 39.
• You divide the denominator (39) by 39.

Thus, \(\frac{78}{39}\) simplifies to \(\frac{2}{1}\), which is simply the whole number 2. Fraction simplification helps in presenting the answer in its most understandable form.