Problem 28 - 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 two fractions, they must have the same denominator. The denominators of \(\frac{9}{10}\) and \(\frac{-11}{8}\) are 10 and 8 respectively. The least common multiple (LCM) of 10 and 8 is 40.

Step 2: Convert Fractions

Convert each fraction to an equivalent fraction with the common denominator 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 that the fractions have the same denominator, add them: \(\frac{36}{40} + \frac{-55}{40} = \frac{36 + (-55)}{40} = \frac{-19}{40}\).

Step 4: Simplify the Result

The fraction \(\frac{-19}{40}\) is already in its simplest form as 19 and 40 have no common factors other than 1.

Key Concepts

least common multiple

When working with fractions, finding a common denominator is essential for addition or subtraction. The easiest way to find a common denominator is to use the least common multiple (LCM) of the denominators. The LCM of two numbers is the smallest number that is a multiple of both. For example, the denominators in the problem are 10 and 8.

To find the LCM of 10 and 8, list their multiples:
- Multiples of 10: 10, 20, 30, 40, 50, 60, ...
- Multiples of 8: 8, 16, 24, 32, 40, 48, ...
The smallest common multiple is 40. This becomes our common denominator.

Using the LCM ensures that you can seamlessly continue with the operation. Now, with the common denominator established, you can proceed to convert the original fractions to equivalent ones with this new denominator.

equivalent fractions

Equivalent fractions are different fractions that represent the same value. They can be found by multiplying or dividing the numerator and the denominator of a fraction by the same number. In our problem, we needed to convert \(\frac{9}{10}\) and \(\frac{-11}{8}\) to fractions with the common denominator 40.

For \(\(\frac{9}{10}\)\), we multiply both the numerator and the denominator by 4 (since \(10 * 4 = 40\)):\(\(\frac{9 * 4}{10 * 4} = \frac{36}{40}\)\). For \(\(\frac{-11}{8}\)\), we do similarly, but this time multiply both the numerator and the denominator by 5 (since \(8 * 5 = 40\)):\(\(\frac{-11 * 5}{8 * 5} = \frac{-55}{40}\)\).

Now, both fractions are \(\(\frac{36}{40}\)\) and \(\(\frac{-55}{40}\)\), making them easier to add together. This step is crucial because it allows for direct operations on the fractions.

simplifying fractions

Simplifying fractions means reducing them to their simplest form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD). However, sometimes a fraction is already in simplest form if the GCD is 1.

After adding \(\(\frac{36}{40}\)\) and \(\(\frac{-55}{40}\)\), we got \(\(\frac{-19}{40}\)\). We need to check if this fraction can be simplified further.
- The absolute value of the numerator is 19.
- The denominator is 40.

Since 19 is a prime number, it has no divisors other than 1 and itself. Check if 19 and 40 have any common factors other than 1 by attempting to divide 40 by 19. The result isn’t an integer, confirming they share no common factors.

Thus, \(\(\frac{-19}{40}\)\) is already in its simplest form, and the problem is solved.