The value of `(1)/(20)+(1)/(30)+(1)/(42)+(1)/(56)+(1)/(72)+(1)/(90)` is

Options

A. `(1)/(10)`
B. `(3)/(5)`
C. `(3)/(20)`
D. `(7)/(20)`

Answer

To find the value of the expression \( \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} \), we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
To add the fractions, we need a common denominator. The first step is to find the LCM of the denominators: 20, 30, 42, 56, 72, and 90.

- **Prime Factorization**:
- \( 20 = 2^2 \times 5 \)
- \( 30 = 2 \times 3 \times 5 \)
- \( 42 = 2 \times 3 \times 7 \)
- \( 56 = 2^3 \times 7 \)
- \( 72 = 2^3 \times 3^2 \)
- \( 90 = 2 \times 3^2 \times 5 \)

- **Taking the highest power of each prime**:
- \( 2^3 \) from 56
- \( 3^2 \) from 72
- \( 5^1 \) from 20 or 30
- \( 7^1 \) from 42 or 56

Thus, the LCM is:
\[ LCM = 2^3 \times 3^2 \times 5^1 \times 7^1 = 8 \times 9 \times 5 \times 7 = 2520 \]

### Step 2: Rewrite each fraction with the common denominator
Now we convert each fraction to have the denominator of 2520:

- \( \frac{1}{20} = \frac{126}{2520} \) (since \( 2520 \div 20 = 126 \))
- \( \frac{1}{30} = \frac{84}{2520} \) (since \( 2520 \div 30 = 84 \))
- \( \frac{1}{42} = \frac{60}{2520} \) (since \( 2520 \div 42 = 60 \))
- \( \frac{1}{56} = \frac{45}{2520} \) (since \( 2520 \div 56 = 45 \))
- \( \frac{1}{72} = \frac{35}{2520} \) (since \( 2520 \div 72 = 35 \))
- \( \frac{1}{90} = \frac{28}{2520} \) (since \( 2520 \div 90 = 28 \))

### Step 3: Add the fractions
Now we can add the fractions together:
\[ \frac{126}{2520} + \frac{84}{2520} + \frac{60}{2520} + \frac{45}{2520} + \frac{35}{2520} + \frac{28}{2520} \]

Combine the numerators:
\[ 126 + 84 + 60 + 45 + 35 + 28 = 378 \]

So, we have:
\[ \frac{378}{2520} \]

### Step 4: Simplify the fraction
Now we simplify \( \frac{378}{2520} \):
- Find the GCD of 378 and 2520. The GCD is 126.
- Divide both the numerator and the denominator by 126:
\[ \frac{378 \div 126}{2520 \div 126} = \frac{3}{20} \]

### Final Answer
Thus, the value of \( \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} \) is \( \frac{3}{20} \).

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