Adding Fractions with Unlike Denominators: 4/10 + 5/12

To solve this problem, we need to add the fractions $\frac{4}{10}$ and $\frac{5}{12}$ by first finding a common denominator.

Step 1: Find the Least Common Denominator (LCD)

The denominators are $10$ and $12$. The least common multiple of $10$ and $12$ can be determined by prime factorization:

• $10 = 2 \times 5$
• $12 = 2^2 \times 3$

For the LCM, take the highest power of each prime:

• For $2$, take $2^2$
• For $3$, take $3$
• For $5$, take $5$

The LCM is $2^2 \times 3 \times 5 = 60$. Thus, the common denominator is $60$.

Step 2: Convert Fractions to Have the Common Denominator

• Convert $\frac{4}{10}$ to $\frac{?}{60}$.

$\frac{4}{10} = \frac{4 \times 6}{10 \times 6} = \frac{24}{60}$.
• Convert $\frac{5}{12}$ to $\frac{?}{60}$.

$\frac{5}{12} = \frac{5 \times 5}{12 \times 5} = \frac{25}{60}$.

Step 3: Add the Fractions

Add $\frac{24}{60}$ and $\frac{25}{60}$:

$\frac{24}{60} + \frac{25}{60} = \frac{24 + 25}{60} = \frac{49}{60}$.

Therefore, the solution to the problem is $\frac{49}{60}$.