Problem 77 - Prealgebra

Question

Simplify. $$\frac{\frac{3}{5}}{\frac{4}{5}}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{3}{4} \).

Step 1: Understand the Expression

The expression we need to simplify is a complex fraction: \( \frac{\frac{3}{5}}{\frac{4}{5}} \). It consists of a fraction in the numerator (\( \frac{3}{5} \)) and a fraction in the denominator (\( \frac{4}{5} \)).

Step 2: Recognize Fraction Division

In a complex fraction where both numerator and denominator are fractions, you can rewrite it as division of those fractions. In mathematics, dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 3: Rewrite as Multiplication

Rewrite the division of the fractions as a multiplication by the reciprocal. Thus, \( \frac{\frac{3}{5}}{\frac{4}{5}} \) becomes \( \frac{3}{5} \times \frac{5}{4} \).

Step 4: Simplify the Multiplication

To simplify \( \frac{3}{5} \times \frac{5}{4} \), multiply the numerators together and the denominators together: \( \frac{3 \times 5}{5 \times 4} \). This yields \( \frac{15}{20} \).

Step 5: Reduce the Fraction

To reduce \( \frac{15}{20} \), find the greatest common divisor (GCD) of the numerator and the denominator, which is 5. Divide both by 5: \( \frac{15 \div 5}{20 \div 5} = \frac{3}{4} \).

Key Concepts

Complex Fractions

A complex fraction is simply a fraction where either the numerator, the denominator, or both are also fractions. This means you see fractions within a fraction! It can look a little overwhelming at first, but breaking it down into steps makes it much easier to understand.
For example, consider the following complex fraction: \( \frac{\frac{3}{5}}{\frac{4}{5}} \). Here, \( \frac{3}{5} \) is the numerator and \( \frac{4}{5} \) is the denominator.
Don't worry if complex fractions seem intimidating at first. Once you recognize that each part is just a simple fraction, you have already won half the battle.

Fraction Division

Fraction division might seem tricky, but there's a neat trick that makes it simpler: multiply by the reciprocal. Whenever you see a division problem involving fractions, you can turn it into multiplication by flipping the second fraction.
For instance, dividing by \( \frac{4}{5} \) is the same as multiplying by the reciprocal of \( \frac{4}{5} \), which is \( \frac{5}{4} \).

• In the problem \( \frac{\frac{3}{5}}{\frac{4}{5}} \), initially, it looks like we're dividing.
• But you can actually rewrite it as \( \frac{3}{5} \times \frac{5}{4} \).

Understanding this step allows you to approach the problem with more confidence.

Reciprocal

The concept of a reciprocal is important in math, especially when dealing with division of fractions. To find the reciprocal of a fraction, you simply swap the numerator and the denominator.
This means if you start with a fraction like \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
Using reciprocals makes division simpler because it turns the operation into multiplication, which is generally easier to perform. For example, in \( \frac{\frac{3}{5}}{\frac{4}{5}} \), recognizing the reciprocal of \( \frac{4}{5} \) is \( \frac{5}{4} \) lets you convert the division into a multiplication task
Always remember: flipping the fraction turns division into multiplication, making calculations straightforward.

Greatest Common Divisor (GCD)

Reducing fractions is an essential part of simplifying math problems, and the greatest common divisor (GCD) plays a crucial role in this process. The GCD is the largest number that divides both the numerator and the denominator without leaving any remainder.

• To simplify \( \frac{15}{20} \), we first need to find their GCD.
• Both 15 and 20 can be divided perfectly by 5.

This means the GCD of 15 and 20 is 5. Once we find this number, we divide both the numerator and the denominator by it:
Calculate: \( \frac{15 \div 5}{20 \div 5} \) to get \( \frac{3}{4} \).
By doing this, we simplify the fraction to its lowest terms, ensuring it remains equivalent to the original fraction.