Problem 29 - Beginning and Intermediate Algebra

Question

Find each sum. $$ \frac{9}{10}+\left(-\frac{11}{8}\right) $$

Step-by-Step Solution

Answer

\(\frac{-19}{40}\)

Step 1: Find a common denominator

To add the fractions \(\frac{9}{10}\) and \(-\frac{11}{8}\), calculate the least common multiple (LCM) of the denominators 10 and 8. The LCM of 10 and 8 is 40.

Step 2: Convert fractions to have the same denominator

Rewrite each fraction with the common denominator of 40. \(\frac{9}{10} = \frac{9 \times 4}{10 \times 4} = \frac{36}{40}\) and \(-\frac{11}{8} = -\frac{11 \times 5}{8 \times 5} = -\frac{55}{40}\).

Step 3: Add the fractions

Now add the two fractions with the common denominator: \(\frac{36}{40} + \frac{-55}{40}\), resulting in \(\frac{36 - 55}{40} = \frac{-19}{40}\).

Step 4: Simplify if necessary

The fraction \(\frac{-19}{40}\) is already in its simplest form.

Key Concepts

common denominator

When adding fractions, the first essential step is to find a common denominator. You can only combine fractions directly if they share the same denominator. The common denominator is a number that both denominators (the bottom parts of the fractions) can evenly divide into.

For example, when adding \(\frac{9}{10}\) and \(-\frac{11}{8}\), the denominators are 10 and 8. Since these fractions have different denominators, we need to find a common one. This shared number will help us rewrite the fractions to make them easier to add.

By finding the common denominator, you ensure both fractions are divided into equal parts, making the addition process straightforward and accurate.

least common multiple

To find our common denominator, we use the concept of the least common multiple (LCM). The LCM of two numbers is the smallest number that both original denominators can divide without leaving a remainder.

For 10 and 8, their LCM is 40. Here’s how we find it:

• List the multiples of each number.
• Multiples of 10: 10, 20, 30, 40, 50,...
• Multiples of 8: 8, 16, 24, 32, 40, 48,...
• The smallest common multiple between these sets is 40.

With 40 as the common denominator, we now rewrite our fractions: \[ \frac{9}{10} = \frac{9 \times 4}{10 \times 4} = \frac{36}{40} \] and \[ -\frac{11}{8} = -\frac{11 \times 5}{8 \times 5} = -\frac{55}{40} \]. Now both fractions have the same denominator and can be added easily.

fraction simplification

In the final addition step, after having converted fractions to a common denominator, we need to add them and simplify the result if necessary. Simplifying means making the fraction as simple as possible by reducing it.

Adding the fractions we have: \[ \frac{36}{40} + \frac{-55}{40} = \frac{36 - 55}{40} = \frac{-19}{40} \].

In this case, the resulting fraction \(\frac{-19}{40}\) is already in its simplest form because there are no common factors between 19 and 40 other than 1. If any common factors existed, we would divide both the numerator (top part) and the denominator (bottom part) by that factor to simplify the fraction.

Simplifying fractions makes them easier to understand and compare. Always check if a fraction can be simplified as a final step in working with fractions.