Problem 1 - Prealgebra

Question

Find each sum. Write in simplest form. $$\frac{1}{15}+\frac{3}{5}$$

Step-by-Step Solution

Answer

The sum is \(\frac{2}{3}\).

Step 1: Find the Least Common Denominator (LCD)

To add fractions, they need a common denominator. The denominators here are 15 and 5. The least common multiple of 15 and 5 is 15. Thus, the least common denominator is 15.

Step 2: Adjust the Fractions

The fraction \(\frac{1}{15}\) already has 15 as a denominator, so it remains the same. For the fraction \(\frac{3}{5}\), multiply the numerator and the denominator by 3 to convert it to \(\frac{9}{15}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, you can add them: \(\frac{1}{15} + \frac{9}{15} = \frac{10}{15}\).

Step 4: Simplify the Sum

The fraction \(\frac{10}{15}\) can be simplified by dividing both the numerator and the denominator by their greatest common factor, which is 5. Dividing both by 5, we get \(\frac{2}{3}\).

Key Concepts

Understanding the Least Common Denominator (LCD)

Adding fractions necessitates a common denominator, ensuring that both fractions are expressed in terms of the same whole. When working with fractions like \( \frac{1}{15} \) and \( \frac{3}{5} \), identify the least common multiple (LCM) of the denominators. For 15 and 5, the LCM is 15, making it the least common denominator (LCD).

To find the LCM, consider the multiples of each number:

• Multiples of 15: 15, 30, 45, ...
• Multiples of 5: 5, 10, 15, 20, ...

The smallest multiple common to both lists is 15, which is why our LCD in this exercise is 15. Using the LCD allows us to rewrite fractions to facilitate addition without changing their value.

Simplifying Fractions to Their Core

Simplifying a fraction means reducing it to its simplest form. This involves dividing the numerator and the denominator by their greatest common factor (GCF). The GCF is the largest number that divides both the numerator and the denominator without leaving a remainder.

For example, the fraction \( \frac{10}{15} \) can be simplified. We first determine the GCF of 10 and 15, which is 5. By dividing both the numerator and the denominator by 5, \( \frac{10}{15} \) reduces to \( \frac{2}{3} \).

Why simplify? It makes answers cleaner and easier to work with. It also provides a universal answer, as this simple form is the same regardless of how the fraction was initially presented.

Adding Fractions with Different Denominators

When fractions do not initially share a common denominator, these steps help align them before they can be added. First, find the least common denominator (LCD) as discussed earlier, then adjust each fraction accordingly.

Taking \( \frac{1}{15} \) and \( \frac{3}{5} \), the LCD is 15. The fraction \( \frac{1}{15} \) already fits this denominator, so no changes are needed. However, for \( \frac{3}{5} \), convert it by multiplying both the numerator and the denominator by 3, resulting in \( \frac{9}{15} \).

Once adjusted, you can simply add the numerators together while keeping the denominator constant:
\[\frac{1}{15} + \frac{9}{15} = \frac{10}{15}\]

Simplify the result, if possible, to maintain neat and consistent solutions. For \( \frac{10}{15} \), divide by the GCF of 5 to achieve the final simplified sum: \( \frac{2}{3} \). This method ensures that students gain a thorough understanding of both operation and value.