Problem 127 - Elementary and Intermediate Algebra

Question

Find \(\frac{1}{2}+\frac{1}{4}+\frac{1}{3}\). Answer in decimal form.

Step-by-Step Solution

Answer

The sum is approximately 1.0833.

Step 1: Finding a Common Denominator

To add fractions, they need to have the same denominator. The fractions given are \( \frac{1}{2} \), \( \frac{1}{4} \), and \( \frac{1}{3} \). The least common denominator (LCD) of 2, 4, and 3 is 12.

Step 2: Convert Fractions to Common Denominator

Convert each fraction to an equivalent fraction with a denominator of 12. \( \frac{1}{2} = \frac{6}{12} \), because \(1 \times 6 = 6\) and \(2 \times 6 = 12\).\( \frac{1}{4} = \frac{3}{12} \), because \(1 \times 3 = 3\) and \(4 \times 3 = 12\).\( \frac{1}{3} = \frac{4}{12} \), because \(1 \times 4 = 4\) and \(3 \times 4 = 12\).

Step 3: Add the Fractions

Now that all fractions have the same denominator, add their numerators: \( \frac{6}{12} + \frac{3}{12} + \frac{4}{12} = \frac{13}{12} \).

Step 4: Convert to Decimal Form

Divide the numerator by the denominator to get the decimal form: \( \frac{13}{12} = 1.0833\) (rounded to four decimal places).

Key Concepts

Common Denominator

When adding fractions, it's crucial to align their denominators. A *common denominator* is like a shared base for fractions, enabling direct addition. Imagine trying to add quarters to dollars; without the same units, it's confusing. Similarly, fractions need the same denominator to add easily.
For instance, to add \( \frac{1}{2} \), \( \frac{1}{4} \), and \( \frac{1}{3} \), we find a common denominator for 2, 4, and 3. Here, the *least common denominator* comes into play, which is the smallest number divisible by each denominator. This simplifies the process and avoids unnecessary large numbers.

Least Common Denominator

The *least common denominator* (LCD) is pivotal for adding fractions. It is the smallest number that all denominators can divide into evenly. Discovering the LCD ensures minimal adjustments, making our calculations simpler and faster.
To find the LCD for \( \frac{1}{2} \), \( \frac{1}{4} \), and \( \frac{1}{3} \), list the multiples of each denominator:

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

The first common multiple is 12, making it our LCD. Once you have the LCD, convert each fraction to this new denominator so they share the same base. This enables seamless addition of the numerators, completing the addition process.

Convert Fractions to Decimal

Converting fractions to decimals is the culmination of the addition process. Once you have a single fraction, dividing its numerator by its denominator transforms it into decimal form, which is often more intuitive to understand and compare.
After adding your fractions and simplifying them into one fraction, say \( \frac{13}{12} \) in this case, you take the numerator 13 and divide it by the denominator 12. This simple division gives a decimal, 1.0833 when rounded to four decimal places.
Decimals provide a straightforward way to represent fractions in a standard numerical form, making them especially useful in various applications like calculations, measurements, and everyday situations.