Problem 77 - Prealgebra

Question

Add or subtract as indicated. $$\frac{2}{7}+\frac{1}{14}$$

Step-by-Step Solution

Answer

\( \frac{5}{14} \)

Step 1: Find a Common Denominator

To add fractions, they need to have the same denominator. The denominators here are 7 and 14. The least common multiple of 7 and 14 is 14. So, we will convert both fractions to have a denominator of 14.

Step 2: Convert the First Fraction

Convert \( \frac{2}{7} \) to a fraction with a denominator of 14. Multiply both the numerator and the denominator by 2: \( \frac{2}{7} \times \frac{2}{2} = \frac{4}{14} \).

Step 3: Align the Second Fraction with the Common Denominator

The second fraction is already \( \frac{1}{14} \), so it remains unchanged as both fractions now have a common denominator of 14.

Step 4: Add the Fractions

With the fractions having a common denominator, add the numerators: \( \frac{4}{14} + \frac{1}{14} = \frac{4 + 1}{14} = \frac{5}{14} \).

Step 5: Final Result

The result of adding the fractions \( \frac{2}{7} \) and \( \frac{1}{14} \) is \( \frac{5}{14} \).

Key Concepts

Least Common Multiple

Before you can add fractions with unlike denominators, a crucial first step is to identify the least common multiple (LCM) of the denominators involved. The LCM is the smallest number that both denominators can divide evenly into. Consider it as finding a common stage where both fractions can perform together.

To find the LCM of 7 and 14, begin with listing the multiples of each number:

• Multiples of 7: 7, 14, 21, 28, ...
• Multiples of 14: 14, 28, 42, ...

You'll notice that 14 is the smallest number that appears in both lists. Therefore, 14 is the least common multiple of 7 and 14. The LCM helps adjust fractions so they share a common denominator, setting the stage for addition or subtraction.

Finding a Common Denominator

Now that you've identified the LCM, it's time to convert each fraction so they share this common denominator. This process is like updating both fractions to a common language, making them easier to work with.

The task here involves tuning fractions so they align perfectly under this common base. For our example:

• For the fraction \( \frac{2}{7} \): Multiply both the numerator and denominator by 2 (because \(7 \times 2 = 14\), our LCM). So, \( \frac{2}{7} \times \frac{2}{2} = \frac{4}{14} \).
• The fraction \( \frac{1}{14} \) already matches the common denominator of 14 since it was very accommodating from the start. So, it remains \( \frac{1}{14} \).

With both fractions now sharing the denominator 14, you're ready for the simple operation of adding them.

Simplifying Fractions

After aligning the fractions with a common denominator and performing the addition, the resulting fraction might need simplification. Simplification means expressing the fraction in its simplest or most reduced form by ensuring the numerator and denominator have no common factors other than 1.

In our resolved fraction \( \frac{5}{14} \), it's important to check if it can be reduced further. Examine the greatest common factor (GCF) of the numerator and denominator.

• The number 5 is a prime number, and 14 can only be divided by 1, 2, 7, or itself—14. This means their greatest common factor is 1.

Since 5 and 14 share no common factors other than 1, \( \frac{5}{14} \) is already in simplest form. Therefore, this fraction represents your final simplified answer.