Problem 29 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{4}{5}-\frac{1}{10} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{7}{10} \).

Step 1: Identify the problem

We need to subtract two fractions: \( \frac{4}{5} \) and \( \frac{1}{10} \).

Step 2: Find a common denominator

To subtract fractions, they need to have the same denominator. The denominators here are 5 and 10. The least common multiple of 5 and 10 is 10.

Step 3: Adjust fractions to have common denominators

Convert \( \frac{4}{5} \) to a fraction with a denominator of 10. Multiply both the numerator and the denominator by 2: \[ \frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10} \].

Step 4: Subtract the fractions

Subtract the second fraction from the first by subtracting the numerators: \[ \frac{8}{10} - \frac{1}{10} = \frac{8 - 1}{10} = \frac{7}{10} \].

Step 5: Simplify the result

The fraction \( \frac{7}{10} \) is already in its simplest form. Therefore, no further simplification is needed.

Key Concepts

Least Common Denominator

When subtracting fractions, it's essential to have a common denominator. This makes the fractions easier to work with. To find this common denominator, we need the Least Common Denominator (LCD). It's essentially the least common multiple (LCM) of the denominators involved.

For example, in the exercise, we subtract \( \frac{4}{5} \) from \( \frac{1}{10} \). The denominators are 5 and 10. The smallest number that both 5 and 10 can divide into evenly is 10. Thus, the LCD is 10. Once we have this common denominator, we adjust the fractions so they can be easily subtracted.

Identifying the LCD is crucial in fraction operations to ensure we are working with equivalent values, making subtraction straightforward.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. This process helps in making further calculations easier and the results more interpretable.

After subtracting fractions and obtaining a result like \( \frac{7}{10} \), we always check if it can be simplified. A fraction is simplified when the greatest common divisor (GCD) of the numerator and the denominator is 1. In this example, 7 and 10 have no common divisors other than 1, confirming that \( \frac{7}{10} \) is already in its simplest form.

Knowing how to simplify fractions is handy especially in longer computations, making the final result cleaner and easier to work with.

Numerator and Denominator

Fractions consist of two parts — the numerator and the denominator. Understanding these terms is crucial in fraction calculations. The numerator is the top part of a fraction, indicating how many parts of the whole are taken. The denominator is the bottom part, showing into how many equal parts the whole is divided.

In our specific exercise, \( \frac{4}{5} \) has 4 as the numerator, representing 4 parts of a whole divided into 5. Similarly, \( \frac{1}{10} \) represents 1 part of a whole divided into 10 parts.

When subtracting fractions like these, we convert both to have the same denominator. This changes the numerators without affecting the values they represent, allowing us to easily perform arithmetic operations on them.