Problem 46 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ -\frac{5}{6}+\frac{3}{8} $$

Step-by-Step Solution

Answer

The result of the operation is \(-\frac{11}{24}\).

Step 1: Understanding the Operation

We need to add two fractions: \(-\frac{5}{6} + \frac{3}{8}\). The fractions have different denominators, so we cannot add them directly.

Step 2: Finding a Common Denominator

To add fractions, they must have a common denominator. The least common multiple (LCM) of 6 and 8 is 24. We will convert both fractions to have this common denominator.

Step 3: Converting to Equivalent Fractions

Convert each fraction so that the denominator is 24:\(-\frac{5}{6} = -\frac{5 \times 4}{6 \times 4} = -\frac{20}{24}\)\(\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}\)

Step 4: Performing the Addition

Now add the fractions with the common denominator:\(-\frac{20}{24} + \frac{9}{24} = \frac{-20 + 9}{24} = \frac{-11}{24}\)

Step 5: Simplifying the Result (if necessary)

The fraction \(\frac{-11}{24}\) is already in its simplest form because 11 and 24 have no common factors other than 1.

Key Concepts

Adding Fractions

When adding fractions, it's essential that the fractions share the same denominator. This is because the denominator determines the size of the fraction pieces. If the denominators are different, you can't directly add the fractions.
To illustrate, think of \(-\frac{5}{6}\)and\(\frac{3}{8}\)as two different sets of pie pieces. The pie pieces from each fraction are a different size, so we must first convert them into uniform pieces before combining.

• Check if the fractions have the same denominator.
• If not, find a common denominator to proceed with the addition.

Once a common denominator is found, you can easily add the numerators together, keeping the common denominator the same.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. Finding the LCM is crucial when adding fractions with different denominators because it helps convert those fractions to an equivalent form with a common denominator.
For the fractions \(-\frac{5}{6}\) and \(\frac{3}{8}\), the denominators are 6 and 8. We need to find their LCM:

• List the multiples of 6: 6, 12, 18, 24, 30...
• List the multiples of 8: 8, 16, 24, 32...
• Identify the smallest common multiple: 24

Now that we have found the LCM, we can use it to convert our fractions to ones with this shared denominator.

Equivalent Fractions

Once you've determined the least common multiple, you can create equivalent fractions, which retain the same value but have a different denominator.
To convert \(-\frac{5}{6}\) and \(\frac{3}{8}\) to equivalent fractions with the denominator of 24, follow these steps:

• Multiply both the numerator and the denominator of \(-\frac{5}{6}\) by 4 to get \(-\frac{20}{24}\).
• Multiply both the numerator and the denominator of \(\frac{3}{8}\) by 3 to get \(\frac{9}{24}\).

Now both fractions share the same denominator of 24, making addition possible.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves dividing the numerator and the denominator by their greatest common factor (GCF). Alternatively, if a fraction is already in its simplest form, there will be no need for further simplification.
In our operation, after adding equivalent fractions, we obtained \(\frac{-11}{24}\). Here:

• 11 is a prime number, meaning its only divisors are 1 and 11.
• 24 has several divisors: 1, 2, 3, 4, 6, 8, 12, and 24.

Because the only common factor between 11 and 24 is 1, this fraction is already as simple as it can be.