Problem 27 - Beginning and Intermediate Algebra

Question

Find each sum. $$ \frac{5}{8}+\left(-\frac{17}{12}\right) $$

Step-by-Step Solution

Answer

\(\frac{-19}{24}\)

Step 1: Find a common denominator

The denominators are 8 and 12. The least common multiple (LCM) of 8 and 12 is 24. We need to rewrite each fraction with 24 as the denominator.

Step 2: Rewrite the fractions

To convert \(\frac{5}{8}\) to a fraction with a denominator of 24, multiply both the numerator and the denominator by 3: \(\frac{5 \times 3}{8 \times 3} = \frac{15}{24}\). To convert \(\frac{-17}{12}\) to a fraction with a denominator of 24, multiply both the numerator and the denominator by 2: \(\frac{-17 \times 2}{12 \times 2} = \frac{-34}{24}\).

Step 3: Add the fractions

Now that both fractions have the same denominator, add the numerators: \( \frac{15}{24} + \frac{-34}{24} = \frac{15 - 34}{24} \). This simplifies to \( \frac{-19}{24} \).

Step 4: Simplify the result (if necessary)

Check if \( \frac{-19}{24} \) can be simplified further. Since 19 and 24 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

least common multiple

When adding fractions, it's crucial to have a common denominator. This is where the Least Common Multiple (LCM) comes in handy. The LCM of two numbers is the smallest number that both can divide into without any remainder. In our exercise, the denominators are 8 and 12. We need to find the LCM of these two numbers so we can add the fractions properly. Here's how to find the LCM:

• List the multiples of the larger number until you find one that's also a multiple of the smaller number.
• For example, the multiples of 12 are 12, 24, 36, 48, and so on.
• Check these multiples against 8. The first multiple of 12 that is also a multiple of 8 is 24.

Therefore, the LCM of 8 and 12 is 24. Now, we can rewrite each fraction using this common denominator, making it easier to add them together. Understanding the LCM is vital for working with fractions, so practice finding it with different pairs of numbers.

simplifying fractions

To simplify a fraction means to make it as simple as possible - that is, the numerator (top number) and the denominator (bottom number) should have no common factors other than 1. Simplifying makes fractions easier to read and understand.
Here's a step-by-step guide to simplifying fractions:

• Find the Greatest Common Divisor (GCD) of the numerator and the denominator. This is the largest number that divides both without a remainder.
• Divide both the numerator and the denominator by their GCD.

For example, in the fraction \(\frac{15}{24}\), the GCD of 15 and 24 is 3. Divide both 15 and 24 by 3, resulting in \(\frac{5}{8}\). Notice how \(\frac{5}{8}\) is simpler and easier to understand than \(\frac{15}{24}\).
In our example, after adding the fractions, we ended up with \(\frac{-19}{24}\). Since 19 and 24 have no common factors other than 1, the fraction is already in its simplest form.

negative numbers

Negative numbers are numbers less than zero, and they play an essential role in various mathematical operations. When adding fractions, you may come across negative numbers, which can complicate things a bit. In our exercise, we have a negative fraction, \(\frac{-17}{12}\). Here's how to handle them:

• First, treat the numerator separately: -17 in this case.
• Then, perform the same operations as you would for a positive fraction.
• Finally, keep the negative sign with the numerator.

In our example, after converting both fractions to have the denominator of 24, we had \(\frac{15}{24} + \frac{-34}{24}\). The next step involves simply adding the numerators while keeping the denominator the same: 15 - 34 = -19, resulting in \(\frac{-19}{24}\).

Always remember, the negative sign only affects the numerator or the whole fraction, not the denominator. This understanding will help you confidently tackle problems involving negative fractions.