Problem 95 - Elementary and Intermediate Algebra

Question

Divide: \(5 \frac{2}{3} \div 2 \frac{5}{9}\)

Step-by-Step Solution

Answer

The result of the division is \(2 \frac{5}{23}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For \(5 \frac{2}{3}\), multiply the whole number \(5\) by the denominator \(3\) and add the numerator \(2\) to get \(17\), so \(5 \frac{2}{3} = \frac{17}{3}\). For \(2 \frac{5}{9}\), multiply \(2\) by \(9\) and add \(5\) to get \(23\), so \(2 \frac{5}{9} = \frac{23}{9}\).

Step 2: Rewrite the Division as Multiplication

To divide by a fraction, you multiply by its reciprocal. Change the division \(\frac{17}{3} \div \frac{23}{9}\) to multiplication by flipping the second fraction: \(\frac{17}{3} \times \frac{9}{23}\).

Step 3: Multiply the Fractions

Multiply the numerators and the denominators: \(17 \times 9 = 153\) and \(3 \times 23 = 69\). Thus, you get \(\frac{153}{69}\).

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{153}{69}\) by finding the greatest common divisor (GCD) of \(153\) and \(69\). The GCD is \(3\). Divide both the numerator and the denominator by \(3\): \(\frac{153 \div 3}{69 \div 3} = \frac{51}{23}\).

Step 5: Convert Improper Fraction to Mixed Number

Since \(\frac{51}{23}\) is an improper fraction, convert it to a mixed number. Divide \(51\) by \(23\), which goes \(2\) times with a remainder of \(5\). So, \(\frac{51}{23} = 2 \frac{5}{23}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. These are generally easier to understand and visualize since they express both parts separately. For example, in the mixed number \(5 \frac{2}{3}\), the whole number is \(5\) and the fraction is \(\frac{2}{3}\).
To work with or manipulate mixed numbers mathematically, they must first be converted into improper fractions. This conversion simplifies operations like addition, subtraction, multiplication, and division involving fractions.

• To convert a mixed number into an improper fraction: multiply the whole number by the denominator of the fraction and add the numerator. This sum becomes the new numerator, with the original denominator unchanged.

Let's apply this to \(2 \frac{5}{9}\): multiply \(2\) by \(9\) to get \(18\), then add \(5\) to get \(23\). Therefore, \(2 \frac{5}{9}\) converts to \(\frac{23}{9}\).

Improper Fractions

Improper fractions have numerators that are greater than or equal to their denominators. They are useful in calculations as they allow for more straightforward arithmetic operations.
In the division example, \(\frac{17}{3}\) and \(\frac{23}{9}\) are improper fractions. Both represent quantities larger than 1 whole.
When converting back and forth between improper fractions and mixed numbers, it's good to remember:

• If the numerator is larger than the denominator, it can be expressed as a mixed number for easier understanding, but should remain improper for calculations.

This flexibility of expression makes improper fractions practical for all kinds of fraction operations.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This means using the lowest possible numerator and denominator while maintaining the same value.
In the exercise, we simplified \(\frac{153}{69}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which was \(3\).

• To find the GCD, list the factors of both numbers and choose the highest one common to both.
• Divide both the numerator and denominator by this number to simplify the fraction.

The result, \(\frac{51}{23}\), is in its simplest form, revealing the same value but in a clearer and more understandable expression.

Reciprocals

A reciprocal of a fraction is what you multiply the fraction by to get 1. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
In division, dividing by a fraction is the same as multiplying by its reciprocal. This stems from the property that multiplication undoes division and vice versa.

• So, to divide by \(\frac{23}{9}\), multiply by its reciprocal, \(\frac{9}{23}\).

This step is crucial because it ensures the operation complies with the rule of 'invert and multiply,' simplifying division into a more straightforward multiplication task. This approach enhances both accuracy and understanding of fraction division.