`(1)/(20)+(1)/(30)+(1)/(42)+(1)/(56)+(1)/(72)+(1)/(90)+(1)/(110)+(1)/(132)=`?

Options

A. `(1)/(8)`
B. `(1)/(7)`
C. `(1)/(6)`
D. `(1)/(10)`

Answer

To solve the equation

\[ \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} + \frac{1}{132} \]

we will find a common denominator and then sum the fractions.

### Step 1: Find the Least Common Multiple (LCM)

First, we need to find the LCM of the denominators: 20, 30, 42, 56, 72, 90, 110, and 132.

- The prime factorization of each number is:
- \(20 = 2^2 \times 5\)
- \(30 = 2 \times 3 \times 5\)
- \(42 = 2 \times 3 \times 7\)
- \(56 = 2^3 \times 7\)
- \(72 = 2^3 \times 3^2\)
- \(90 = 2 \times 3^2 \times 5\)
- \(110 = 2 \times 5 \times 11\)
- \(132 = 2^2 \times 3 \times 11\)

The LCM will take the highest power of each prime:
- \(2^3\) from 56,
- \(3^2\) from 72,
- \(5^1\) from 20, 30, or 90,
- \(7^1\) from 42 or 56,
- \(11^1\) from 110 or 132.

Thus, the LCM is:

\[ LCM = 2^3 \times 3^2 \times 5^1 \times 7^1 \times 11^1 = 8 \times 9 \times 5 \times 7 \times 11 \]

Calculating this step by step:

1. \(8 \times 9 = 72\)
2. \(72 \times 5 = 360\)
3. \(360 \times 7 = 2520\)
4. \(2520 \times 11 = 27720\)

So, the LCM is \(27720\).

### Step 2: Rewrite each fraction with the common denominator

Now we rewrite each fraction with the common denominator \(27720\):

\[ \frac{1}{20} = \frac{1386}{27720}, \quad \frac{1}{30} = \frac{924}{27720}, \quad \frac{1}{42} = \frac{660}{27720} \]
\[ \frac{1}{56} = \frac{495}{27720}, \quad \frac{1}{72} = \frac{385}{27720}, \quad \frac{1}{90} = \frac{308}{27720} \]
\[ \frac{1}{110} = \frac{252}{27720}, \quad \frac{1}{132} = \frac{210}{27720} \]

### Step 3: Sum the numerators

Now we add the numerators:

\[ 1386 + 924 + 660 + 495 + 385 + 308 + 252 + 210 \]

Calculating this step by step:

1. \(1386 + 924 = 2310\)
2. \(2310 + 660 = 2970\)
3. \(2970 + 495 = 3465\)
4. \(3465 + 385 = 3850\)
5. \(3850 + 308 = 4158\)
6. \(4158 + 252 = 4410\)
7. \(4410 + 210 = 4620\)

So, the sum of the numerators is \(4620\).

### Step 4: Write the final fraction

Now we have:

\[ \frac{4620}{27720} \]

### Step 5: Simplify the fraction

To simplify \(\frac{4620}{27720}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 4620 and 27720 is 4620.

\[ \frac{4620 \div 4620}{27720 \div 4620} = \frac{1}{6} \]

### Final Answer

Thus,

\[ \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} + \frac{1}{132} = \frac{1}{6} \]