What is the value of (1/6) + (1/12) + (1/20)?

Options

A. `2//10`
B. `3//10`
C. `1//2`
D. `2//5`

Answer

To find the value of \( \frac{1}{6} + \frac{1}{12} + \frac{1}{20} \), we will follow these steps:

### Step 1: Identify the denominators
The denominators of the fractions are 6, 12, and 20.

### Step 2: Find the Least Common Multiple (LCM)
To add these fractions, we need a common denominator. We find the LCM of 6, 12, and 20.

- The prime factorization of 6 is \( 2 \times 3 \).
- The prime factorization of 12 is \( 2^2 \times 3 \).
- The prime factorization of 20 is \( 2^2 \times 5 \).

The LCM is found by taking the highest power of each prime:
- For \( 2 \): highest power is \( 2^2 \) (from 12 and 20).
- For \( 3 \): highest power is \( 3^1 \) (from 6 and 12).
- For \( 5 \): highest power is \( 5^1 \) (from 20).

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60. \]

### Step 3: Rewrite each fraction with the common denominator
Now we convert each fraction to have a denominator of 60:
- For \( \frac{1}{6} \):
\[ \frac{1}{6} = \frac{1 \times 10}{6 \times 10} = \frac{10}{60}. \]
- For \( \frac{1}{12} \):
\[ \frac{1}{12} = \frac{1 \times 5}{12 \times 5} = \frac{5}{60}. \]
- For \( \frac{1}{20} \):
\[ \frac{1}{20} = \frac{1 \times 3}{20 \times 3} = \frac{3}{60}. \]

### Step 4: Add the fractions
Now we can add the fractions:
\[ \frac{10}{60} + \frac{5}{60} + \frac{3}{60} = \frac{10 + 5 + 3}{60} = \frac{18}{60}. \]

### Step 5: Simplify the result
Now we simplify \( \frac{18}{60} \):
- The greatest common divisor (GCD) of 18 and 60 is 6.
\[ \frac{18 \div 6}{60 \div 6} = \frac{3}{10}. \]

### Final Answer
Thus, the value of \( \frac{1}{6} + \frac{1}{12} + \frac{1}{20} \) is \( \frac{3}{10} \).

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