Problem 71 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{20}{30}-\frac{2}{3} $$

Step-by-Step Solution

Answer

0

Step 1: Find Common Denominator

First, find a common denominator for the fractions: \ \( \frac{20}{30} \text{ and } \frac{2}{3} \). The least common denominator (LCD) of 30 and 3 is 30. So, you need to express \( \frac{2}{3} \) with a denominator of 30.

Step 2: Convert Fractions

Convert \( \frac{2}{3} \) to a fraction with a denominator of 30: \ \( \frac{2}{3} × \frac{10}{10} = \frac{20}{30} \). Now both fractions have the same denominator.

Step 3: Perform Subtraction

With both fractions having the same denominator, perform the subtraction: \ \( \frac{20}{30} - \frac{20}{30} = \frac{0}{30} \).

Step 4: Simplify the Result

Simplify \( \frac{0}{30} \) to get 0.

Step 5: Verify with a Calculator

Calculate \( \frac{20}{30} - \frac{2}{3} \) using a calculator to verify the result is 0.

Key Concepts

Common Denominator

To subtract fractions, they need to have the same denominator. This is called a common denominator. When fractions have different denominators, it can be difficult to compare or perform operations like addition and subtraction. _Denominators_ are the _bottom_ numbers in fractions, showing how many equal parts the whole is divided into.

For example, when working with \(\frac{20}{30}\) and \(\frac{2}{3}\), we need to make their denominators the same. The first fraction has a denominator of 30, while the second has a denominator of 3. To perform subtraction, we find a common denominator.

Bullet points to remember:

• Fractions need a common denominator for addition and subtraction.
• The denominator is the bottom number in a fraction.
• Finding a common denominator involves identifying a common multiple of the denominators.

This step makes calculations easier and ensures accurate results.

Least Common Denominator

Identifying the least common denominator (LCD) is crucial in fraction subtraction.
The LCD is the smallest common multiple of the denominators of the fractions. Think of it as finding the smallest number that both denominators can divide into without leaving a remainder.

For instance, in the fractions \(\frac{20}{30}\) and \(\frac{2}{3}\), the denominators are 30 and 3. The smallest multiple that 30 and 3 share is 30. So, 30 is the least common denominator. This means we need to convert \(\frac{2}{3}\) to a fraction with a denominator of 30.

Steps to find the LCD:

• List the multiples of each denominator.
• Select the smallest multiple common to both lists.

Using the LCD ensures that calculations are simplified and accurate, making it easier to perform fraction operations like subtraction.

Simplifying Fractions

After performing operations with fractions, it's important to simplify the result. Simplifying a fraction means reducing it to its smallest form. This makes the fraction easier to understand and work with.

Reducing a fraction involves finding the greatest common divisor (GCD) of the numerator and denominator, then dividing both by this number. However, in some cases, like in our example \(\frac{0}{30}\), the result is already simplified. \(\frac{0}{30}\) simplifies directly to 0 because zero divided by any number is zero.

Steps to simplify fractions:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and denominator by the GCD.
• Write down the simplified fraction.

Simplifying ensures that fractions are in their simplest, most understandable form. This helps in further calculations and comparisons.