We need to find the Least Common Denominator (LCD) and the simplified sum of the fractions $\frac{1}{4}$ and $\frac{3}{5}$ using the prime factorization method.

Step 1: Prime Factorization of the Denominators

• Denominator of $\frac{1}{4}$: $4 = 2^2$
• Denominator of $\frac{3}{5}$: $5 = 5^1$

Step 2: Find the LCD

To find the LCD, we take the highest powers of all prime factors involved.

• From $4$, we have $2^2$
• From $5$, we have $5^1$

Thus, the LCD is $2^2 \times 5^1 = 4 \times 5 = 20$.

Step 3: Convert Fractions to have the LCD

We now express both fractions with the denominator 20.

• For $\frac{1}{4}$: Multiply the numerator and denominator by 5: $\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$
• For $\frac{3}{5}$: Multiply the numerator and denominator by 4: $\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}$

Step 4: Add the Fractions

Now add the two fractions with the same denominator: $\frac{5}{20} + \frac{12}{20} = \frac{17}{20}$

Thus, the simplified sum is $\frac{17}{20}$.

Final Answer:

• LCD: 20
• Simplified sum: $\frac{17}{20}$

The correct answer is d. $\frac{17}{20}$.

Would you like more details on any of these steps?

Here are 5 questions to expand on this:

1. How do you perform prime factorization for more complex numbers?
2. Why is it necessary to find the LCD when adding or subtracting fractions?
3. What are some tips for simplifying fractions after performing addition?
4. How does the LCD relate to the Least Common Multiple (LCM)?
5. What happens if fractions have the same denominator from the start?

Tip: Always check if the resulting fraction can be simplified further by finding the greatest common divisor (GCD) of the numerator and denominator!