Problem 91 - Elementary Algebra

Question

For exercises 91-94, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Evaluate: \(\frac{11}{15}+\frac{2}{15}\) Incorrect Answer: \(\frac{11}{15}+\frac{2}{15}\) $$ =\frac{13}{30} $$

Step-by-Step Solution

Answer

\( \frac{13}{15} \)

Step 1: Identify the mistake

Review the given incorrect solution \ \( \frac{11}{15} + \frac{2}{15} = \frac{13}{30} \). The mistake lies in the addition of the fractions. Instead of correctly adding the numerators, the denominators were mistakenly added.

Step 2: Add the fractions correctly

To correctly add the fractions \( \frac{11}{15} \) and \( \frac{2}{15} \), ensure the denominators remain the same, and only add the numerators. \ \[ \frac{11}{15} + \frac{2}{15} = \frac{11 + 2}{15} \]

Step 3: Simplify the result

Simplify the fraction obtained from step 2: \ \[ \frac{11 + 2}{15} = \frac{13}{15} \] So, the correct answer is \( \frac{13}{15} \).

Key Concepts

Common Denominators

When adding fractions, one of the most important steps is to ensure the denominators (the bottom numbers of the fractions) are the same. This is known as having common denominators. Without common denominators, you cannot directly add or subtract the fractions.
For example, in the problem \(\frac{11}{15} + \frac{2}{15}\), both fractions already have the same denominator, which is 15. This makes the addition process straightforward. If the denominators were different, you would need to find a common denominator before adding them. Finding a common denominator usually involves finding the least common multiple (LCM) of the two denominators.
Here’s what you remember:

• Ensure the denominators are the same.
• If not, find the common denominator.
• Only then proceed to add the numerators.

Simplifying Fractions

Simplifying fractions means making the fraction as simple as possible. You do this by dividing the numerator (the top number) and the denominator by their greatest common divisor (GCD). However, in some cases like our exercise, the final fraction is already in its simplest form.
After you have added fractions and obtained a result, you might need to simplify it. For the problem \(\frac{11}{15} + \frac{2}{15} = \frac{13}{15}\), the answer \(\frac{13}{15}\) is already simplified because 13 and 15 have no common factors other than 1. Steps to simplify:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and denominator by the GCD.
• Verify the fraction is in its simplest form.

If the result is already simplified, just confirm that the numbers have no common factors greater than 1.

Numerator Addition

Adding the numerators (the top numbers) of fractions is another key step. This is straightforward but must be done only when the denominators are equal. Failing to keep denominators unchanged, as seen in the incorrect solution \(\frac{11}{15} + \frac{2}{15} = \frac{13}{30}\), leads to errors.
Correctly, you just add the numerators:
\[\frac{11}{15} + \frac{2}{15}\rightarrow \frac{11+2}{15}\]
Which simplifies to: \[\frac{13}{15}\]

• Do not change the denominators.
• Add only the numerators.
• Ensure the final answer is simplified.

By keeping these rules in mind, you will always get the correct result when adding fractions.