Problem 1 - Algebra 2 and Trigonometry

Question

Tony said that \(\frac{3}{1-\frac{1}{5}}\) is irrational because it is not the ratio of integers and is therefore not a rational number. Do you agree with Tony? Explain why or why not.

Step-by-Step Solution

Answer

No, Tony is incorrect; the expression is rational.

Step 1: Simplifying the Denominator

First, let's simplify the denominator of the expression \( \frac{3}{1-\frac{1}{5}} \). The denominator is \( 1 - \frac{1}{5} \). Subtract \( \frac{1}{5} \) from 1 to get \( \frac{5}{5} - \frac{1}{5} = \frac{4}{5} \). So the expression simplifies to \( \frac{3}{\frac{4}{5}} \).

Step 2: Rewriting the Expression

Now, rewrite the expression \( \frac{3}{\frac{4}{5}} \) as a multiplication problem by using the property \( \frac{a}{\frac{b}{c}} = a \cdot \frac{c}{b} \). This gives us \( 3 \cdot \frac{5}{4} \).

Step 3: Simplifying the Expression

Multiply the fraction out to get \( 3 \cdot \frac{5}{4} = \frac{15}{4} \).

Step 4: Determining if the Result is Rational

A rational number is defined as a number that can be expressed as the ratio of two integers. The expression \( \frac{15}{4} \) is a ratio of two integers (15 and 4).

Key Concepts

Irrational Numbers

When talking about numbers, we usually deal with two broad categories, rational and irrational numbers. Irrational numbers are those that cannot be expressed as a simple fraction. This means they cannot be written as the quotient of two integers. Common examples of irrational numbers include \( \pi \) (pi) and \( \sqrt{2} \) (the square root of 2).
These numbers go on infinitely without repeating or terminating, which makes them tricky to pinpoint on the number line.
In contrast, rational numbers can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \). This is important when determining if a number is rational or irrational, such as in the case given in the exercise.
The statement by Tony about the expression \( \frac{3}{1-\frac{1}{5}} \) being irrational needs revisiting because the simplified form \( \frac{15}{4} \) can indeed be represented as a fraction of integers, making this number rational.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple or as small as possible, but equivalent in value.
This involves reducing a fraction by dividing both the numerator (top part of the fraction) and the denominator (bottom part of the fraction) by their greatest common divisor (GCD).
In our example, when we have \( \frac{3}{\frac{4}{5}} \), instead of using a direct simplification method, we use a property to transform it into a multiplication problem.

• Switch from division to multiplication: \( \frac{a}{\frac{b}{c}} = a \times \frac{c}{b} \).
• This property lets us rewrite \( \frac{3}{\frac{4}{5}} \) into \( 3 \times \frac{5}{4} \).

Once in this form, solving it is straightforward; we multiply to get \( \frac{15}{4} \). This fraction is in its simplest form as 15 and 4 have no common factors other than 1.

Properties of Fractions

Fractions have several key properties which are essential for understanding and working with them effectively.
These properties help in simplifying fractions, comparing them, and performing arithmetic operations.

• Reciprocal: This is when you swap the numerator and the denominator of a fraction, transforming \( \frac{a}{b} \) into \( \frac{b}{a} \).
• Multiplication of Fractions: To multiply fractions, simply multiply across the numerators and denominators. For example, \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \).
• Simplification: This reduces the fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor.

In the context of the exercise, knowing how to simplify and manipulate fractions allowed us to properly interpret \( \frac{3}{\frac{4}{5}} \) as \( \frac{15}{4} \). Understanding these properties confirms that \( \frac{15}{4} \) is indeed rational because it’s a simple fraction, consisting of a numerator and a denominator that are both integers.