Problem 46 - Algebra for College Students

Question

Perform the following operations with real numbers. $$\frac{5}{6}-\left(-\frac{2}{9}\right)$$

Step-by-Step Solution

Answer

The result is \( 1 \frac{1}{18} \).

Step 1: Identify the Problem

You need to solve the expression \( \frac{5}{6} - \left(-\frac{2}{9}\right) \). This involves subtracting a negative fraction which is the same as adding the positive value of that fraction.

Step 2: Convert Subtraction to Addition

Rewrite the expression as an addition, since subtracting a negative number is equivalent to adding: \( \frac{5}{6} + \frac{2}{9} \).

Step 3: Find a Common Denominator

Identify the least common denominator for the fractions \( \frac{5}{6} \) and \( \frac{2}{9} \). The least common multiple of 6 and 9 is 18.

Step 4: Convert Fractions

Convert each fraction to an equivalent fraction with a denominator of 18. - \( \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \)- \( \frac{2}{9} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18} \)

Step 5: Add the Fractions

Add the fractions: \( \frac{15}{18} + \frac{4}{18} = \frac{19}{18} \).

Step 6: Simplify the Result

The fraction \( \frac{19}{18} \) is an improper fraction. It simplifies to \( 1 \frac{1}{18} \).

Key Concepts

Fraction Addition

Adding fractions might seem difficult at first, but it's quite straightforward once you understand the steps involved. When dealing with fractions, it is essential to remember that adding or subtracting fractions requires us to manipulate the numerators while considering a common denominator.

• Switch subtraction to addition: For instance, if you see something like \( \frac{5}{6} - \left(-\frac{2}{9}\right) \), remember that subtracting a negative is the same as adding.
• Rewrite the expression as addition: Convert \( \frac{5}{6} - (-\frac{2}{9}) \) to \( \frac{5}{6} + \frac{2}{9} \).

Let's move on to how you can find that common denominator to proceed with your addition.

Common Denominator

To successfully add fractions, finding a common denominator is crucial. The common denominator is simply a shared multiple of both denominators.

• Identify the denominators: In the example \( \frac{5}{6} + \frac{2}{9} \), the denominators are 6 and 9.
• Find the least common multiple (LCM): The smallest common multiple of 6 and 9 is 18. This becomes our new denominator.

Once you have your common denominator, convert each fraction so they both have this new denominator:

• Convert each fraction: \( \frac{5}{6} = \frac{15}{18} \) and \( \frac{2}{9} = \frac{4}{18} \).

Now, you can easily add the fractions because they share a common base.

Improper Fraction

When performing fraction addition, you might end up with an improper fraction as your result. An improper fraction has a numerator larger than or equal to the denominator.

• Add the fractions: With our example, adding \( \frac{15}{18} + \frac{4}{18} = \frac{19}{18} \), we indeed have an improper fraction.
• Simplify: To convert \( \frac{19}{18} \) into a mixed number, you divide the numerator by the denominator. So, \( 19 \div 18 \) gives 1 with a remainder of 1.
• Write the mixed number: Hence, we get 1 whole and \( \frac{1}{18} \), which is \( 1 \frac{1}{18} \).

Simplifying doesn't just make the fraction look nicer; it often makes it easier to use in equations or further calculations.