Problem 32 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{5}{9} \div \frac{5}{6}$$

Step-by-Step Solution

Answer

The value is \( \frac{2}{3} \).

Step 1: Understand the Division of Fractions

When you divide fractions, you multiply the first fraction by the reciprocal of the second fraction. So, for \( \frac{5}{9} \div \frac{5}{6} \), you need to find the reciprocal of \( \frac{5}{6} \).

Step 2: Find the Reciprocal

The reciprocal of a fraction is obtained by flipping its numerator and denominator. Therefore, the reciprocal of \( \frac{5}{6} \) is \( \frac{6}{5} \).

Step 3: Multiply the Fractions

Now multiply \( \frac{5}{9} \) by \( \frac{6}{5} \). The formula for multiplying fractions is \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \). So, \( \frac{5}{9} \times \frac{6}{5} = \frac{5 \times 6}{9 \times 5} \).

Step 4: Simplify the Product

The multiplication gives \( \frac{30}{45} \). To simplify, find the greatest common divisor (GCD) of 30 and 45, which is 15. Divide both the numerator and the denominator by 15: \( \frac{30 \div 15}{45 \div 15} = \frac{2}{3} \).

Key Concepts

Reciprocal of a Fraction

The concept of the reciprocal is crucial in mathematics, especially when working with fractions. A reciprocal is essentially a fraction flipped upside down. This means, for any fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). Reciprocals are important when dividing fractions because they transform the division problem into a multiplication one.
To illustrate with an example, consider the fraction \( \frac{5}{6} \). Its reciprocal is simply \( \frac{6}{5} \), achieved by swapping the numerator and the denominator. Once you have the reciprocal, you can proceed to multiply it by another fraction to solve division problems involving fractions. This method simplifies the process significantly.

Multiplying Fractions

Multiplying fractions is a straightforward process that involves multiplying across the numerators and denominators. Let’s break it down step by step. When given two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), follow this simple formula:

• Multiply the numerators: \( a \times c \)
• Multiply the denominators: \( b \times d \)

This results in the product \( \frac{a \times c}{b \times d} \). For our example of \( \frac{5}{9} \times \frac{6}{5} \), multiply the top numbers (5 and 6) and the bottom numbers (9 and 5). The resulting fraction is \( \frac{30}{45} \).
One key to remember is that multiplying fractions often leads to a fraction that can be simplified, making the process of understanding and working with the result easier.

Simplification of Fractions

Once you've performed arithmetic with fractions, you'll often end up with a fraction that can be simplified. Simplification involves reducing the fraction to its lowest terms, and it's a helpful skill for making fractions easier to work with and understand.
To simplify a fraction \( \frac{m}{n} \), find the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers evenly.

• In our case, the fraction is \( \frac{30}{45} \).
• The GCD of 30 and 45 is 15.
• Divide both the numerator and the denominator by this GCD (15): \( \frac{30 \div 15}{45 \div 15} = \frac{2}{3} \).

The simplified fraction \( \frac{2}{3} \) is easier to interpret and use in further calculations. This simplification process ensures that results are neat and accurate, providing clarity when fractions need to be compared or used in subsequent mathematical operations.