Problem 98 - Elementary Algebra

Question

For exercises \(95-98\), evaluate. $$ \frac{1}{3}+\frac{1}{2}+\frac{1}{4} $$

Step-by-Step Solution

Answer

The sum is \ \ \ \( \frac{13}{12} \ \).

Step 1 - Identify the problem

The goal is to evaluate the sum of three fractions: \ \( \frac{1}{3} + \frac{1}{2} + \frac{1}{4} \ \)

Step 2 - Find a common denominator

Identify the least common multiple (LCM) of the denominators 3, 2, and 4. The LCM of 3, 2, and 4 is 12.

Step 3 - Convert fractions to have the common denominator

Rewrite each fraction with the common denominator of 12: \ \ \ \ \ \ \ \ \ \ \ \( \frac{1}{3} = \frac{4}{12}, \frac{1}{2} = \frac{6}{12}, \frac{1}{4} = \frac{3}{12} \ \)

Step 4 - Add the fractions

Now add the fractions with the same denominator: \ \ \( \frac{4}{12} + \frac{6}{12} + \frac{3}{12} \ \ = \ \ \frac{4+6+3}{12} \ \ = \ \ \frac{13}{12} \ \)

Step 5 - Simplify the fraction if possible

Since \( \frac{13}{12} \ \) is already in its simplest form, we leave it as it is.

Key Concepts

least common multiple

When adding fractions with different denominators, the first step is to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that all the denominators can divide into without leaving a remainder. For the fractions \( \frac{1}{3}, \frac{1}{2}, \frac{1}{4} \), we need a common multiple of 3, 2, and 4.
We list the multiples of each number:

• Multiples of 3: 3, 6, 9, 12, 15, ...
• Multiples of 2: 2, 4, 6, 8, 10, 12, ...
• Multiples of 4: 4, 8, 12, 16, ...

By examining the lists, we see that 12 is the smallest common multiple. Thus, the LCM of 3, 2, and 4 is 12.
With the LCM known, we can proceed to the next steps.

common denominator

A common denominator is essential to add fractions together. This means converting each fraction to an equivalent fraction with the same denominator.
Since the LCM of 3, 2, and 4 is 12, we convert each fraction to have a denominator of 12.
Here's how:

• \( \frac{1}{3} \) becomes \( \frac{4}{12} \) because \( 3 \times 4 = 12 \), so \( \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \).
• \( \frac{1}{2} \) becomes \( \frac{6}{12} \) because \( 2 \times 6 = 12 \), so \( \frac{1 \times 6}{2 \times 6} = \frac{6}{12} \).
• \( \frac{1}{4} \) becomes \( \frac{3}{12} \) because \( 4 \times 3 = 12 \), so \( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \).

All fractions now have a common denominator of 12, making them ready to be added together.

simplifying fractions

Once we have fractions with a common denominator, we can add them together easily.
In this exercise, the converted fractions are \( \frac{4}{12}, \frac{6}{12}, \frac{3}{12} \). By adding the numerators, we get:
\( \frac{4+6+3}{12} = \frac{13}{12} \).
This resulting fraction, \( \frac{13}{12} \), is an improper fraction since the numerator is larger than the denominator.
This fraction is already in its simplest form, as 13 and 12 have no common factors other than 1.
Therefore, the simplified result of adding the given fractions is \( \frac{13}{12} \). Knowing how to simplify fractions ensures the final answer is presented in the cleanest and most reduced form.