Problem 36 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$\frac{2}{5}-\left(-\frac{1}{10}\right)$$

Step-by-Step Solution

Answer

\(\frac{1}{2}\)

Step 1: Rewrite the Problem

Change the operation from subtraction to addition, removing the double negative. The problem becomes: \( \frac{2}{5} + \frac{1}{10} \).

Step 2: Make the Denominators the Same

The first fraction has a denominator of 5, while the second has a denominator of 10. In order for the fractions to be added, they need to have the same denominator. Multiply the numerator and denominator of the first fraction by 2 to get a denominator of 10. The problem becomes: \( \frac{4}{10} + \frac{1}{10} \).

Step 3: Add the Fractions

Now that the fractions have the same denominator, it is easy to add them together. Simply add the numerators and put the result over the common denominator: \( \frac{4+1}{10} = \frac{5}{10} \).

Step 4: Simplify the Fraction

The fraction \( \frac{5}{10} \) can be simplified by dividing both the numerator and the denominator by 5. The fraction simplifies down to \( \frac{1}{2} \).

Key Concepts

Common Denominator

Understanding the concept of a common denominator is crucial when dealing with fraction operations like addition and subtraction. A common denominator is essentially a common multiple of the denominators of the fractions you are working with. When fractions have the same denominator, it allows for straightforward operations because it aligns the point of reference for the fractions. In this exercise, the fractions \(\frac{2}{5}\) and \(\frac{1}{10}\) have different denominators: 5 and 10. To solve the problem, we need to convert these fractions so that they share the same denominator.

• Identify the least common multiple of the denominators. In this case, 10 is the least common multiple of 5 and 10.
• Convert the fraction with the smaller denominator by multiplying both its numerator and denominator to get the common denominator.
• For \(\frac{2}{5}\), multiply the numerator 2 and the denominator 5 by 2, resulting in \(\frac{4}{10}\).

Now both fractions, \(\frac{4}{10}\) and \(\frac{1}{10}\), have a common denominator of 10, which allows them to be easily added together.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This means expressing a fraction in its lowest terms, where the numerator and the denominator have no common factors other than 1. To simplify a fraction, you find the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number. This is crucial because it results in the most simplified version of the fraction, making it easier to understand and work with.

• Consider the fraction \(\frac{5}{10}\).
• Find the GCD of 5 and 10, which is 5.
• Divide both the numerator and the denominator by 5 to simplify the fraction.
• The result is \(\frac{1}{2}\).

Now, \(\frac{1}{2}\) is the simplest form of \(\frac{5}{10}\), making it easier to understand and use in further mathematical operations or real-world applications.

Adding Fractions

Adding fractions involves combining fractions to get a single fraction as a result. To successfully add fractions, they must first have a common denominator. This makes the process of combining them straightforward since only the numerators need to be added while retaining the common denominator. In this exercise, once \(\frac{2}{5}\) is converted to \(\frac{4}{10}\), both fractions have the common denominator necessary for addition.

• Place the fractions to be added: \(\frac{4}{10}\) and \(\frac{1}{10}\).
• Since the denominators are the same, directly add the numerators: 4 + 1 = 5.
• Write the result over the common denominator: \(\frac{5}{10}\).

The fractions have been successfully added, resulting in \(\frac{5}{10}\), which can then be simplified to \(\frac{1}{2}\). This method is straightforward and ensures accuracy when handling fractional additions.