The image shows a mathematical problem involving the addition of fractions:
$$
\frac{1}{4} + \frac{1}{5} + \frac{1}{12}
$$
The task is to solve this problem step by step. Let's break it down:
---
Step 1: Identify the denominators
The denominators of the fractions are:
- \( 4 \)
- \( 5 \)
- \( 12 \)
We need to find the
least common denominator (LCD) of these numbers.
---
Step 2: Prime factorization of the denominators
To find the LCD, we perform prime factorization of each denominator:
- \( 4 = 2 \times 2 \)
- \( 5 = 5 \)
- \( 12 = 2 \times 2 \times 3 \)
---
Step 3: Determine the least common denominator (LCD)
The LCD is found by taking the highest power of each prime number that appears in the factorizations:
- The prime factors involved are \( 2 \), \( 3 \), and \( 5 \).
- From \( 4 = 2^2 \), we take \( 2^2 \).
- From \( 5 = 5^1 \), we take \( 5^1 \).
- From \( 12 = 2^2 \times 3 \), we take \( 2^2 \) and \( 3^1 \).
Thus, the LCD is:
$$
2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60
$$
So, the least common denominator is \( 60 \).
---
Step 4: Rewrite each fraction with the LCD as the denominator
We convert each fraction to an equivalent fraction with a denominator of \( 60 \):
1. For \( \frac{1}{4} \):
$$
\frac{1}{4} = \frac{1 \times 15}{4 \times 15} = \frac{15}{60}
$$
2. For \( \frac{1}{5} \):
$$
\frac{1}{5} = \frac{1 \times 12}{5 \times 12} = \frac{12}{60}
$$
3. For \( \frac{1}{12} \):
$$
\frac{1}{12} = \frac{1 \times 5}{12 \times 5} = \frac{5}{60}
$$
Now, the fractions are:
$$
\frac{15}{60}, \quad \frac{12}{60}, \quad \frac{5}{60}
$$
---
Step 5: Add the fractions
Since the denominators are now the same, we can add the numerators directly:
$$
\frac{15}{60} + \frac{12}{60} + \frac{5}{60} = \frac{15 + 12 + 5}{60} = \frac{32}{60}
$$
---
Step 6: Simplify the resulting fraction
The fraction \( \frac{32}{60} \) can be simplified by finding the greatest common divisor (GCD) of \( 32 \) and \( 60 \).
- The prime factorization of \( 32 \) is \( 2^5 \).
- The prime factorization of \( 60 \) is \( 2^2 \times 3 \times 5 \).
The GCD is \( 2^2 = 4 \).
Divide both the numerator and the denominator by \( 4 \):
$$
\frac{32 \div 4}{60 \div 4} = \frac{8}{15}
$$
---
Final Answer:
The sum of the fractions is:
$$
\boxed{\frac{8}{15}}
$$
Parent Tip: Review the logic above to help your child master the concept of finding lowest common denominator.
