The value of `(3)/(15) +(13)/(14)-(19)/(21) +(31)/(35) -(23)/(30)` is :

Options

A. `(2)/(5)`
B. `(1)/(3)`
C. `(12)/(35)`
D. `(8)/(21)`

Answer

To solve the expression \(\frac{3}{15} + \frac{13}{14} - \frac{19}{21} + \frac{31}{35} - \frac{23}{30}\), we will follow these steps:

### Step 1: Simplify Individual Fractions
First, we can simplify \(\frac{3}{15}\):
\[ \frac{3}{15} = \frac{1}{5} \]
So, the expression becomes:
\[ \frac{1}{5} + \frac{13}{14} - \frac{19}{21} + \frac{31}{35} - \frac{23}{30} \]

### Step 2: Find the LCM of the Denominators
The denominators are \(5, 14, 21, 35, 30\). We need to find the least common multiple (LCM) of these numbers.

- The prime factorization is:
- \(5 = 5^1\)
- \(14 = 2^1 \times 7^1\)
- \(21 = 3^1 \times 7^1\)
- \(35 = 5^1 \times 7^1\)
- \(30 = 2^1 \times 3^1 \times 5^1\)

The LCM will take the highest power of each prime:
\[ \text{LCM} = 2^1 \times 3^1 \times 5^1 \times 7^1 = 210 \]

### Step 3: Convert Each Fraction to Have the Same Denominator
Now, we convert each fraction to have a denominator of \(210\):

- For \(\frac{1}{5}\):
\[ \frac{1}{5} = \frac{1 \times 42}{5 \times 42} = \frac{42}{210} \]

- For \(\frac{13}{14}\):
\[ \frac{13}{14} = \frac{13 \times 15}{14 \times 15} = \frac{195}{210} \]

- For \(\frac{19}{21}\):
\[ \frac{19}{21} = \frac{19 \times 10}{21 \times 10} = \frac{190}{210} \]

- For \(\frac{31}{35}\):
\[ \frac{31}{35} = \frac{31 \times 6}{35 \times 6} = \frac{186}{210} \]

- For \(\frac{23}{30}\):
\[ \frac{23}{30} = \frac{23 \times 7}{30 \times 7} = \frac{161}{210} \]

### Step 4: Substitute Back into the Expression
Now we substitute these back into the expression:
\[ \frac{42}{210} + \frac{195}{210} - \frac{190}{210} + \frac{186}{210} - \frac{161}{210} \]

### Step 5: Combine the Numerators
Combine the numerators over the common denominator:
\[ \frac{42 + 195 - 190 + 186 - 161}{210} \]

Calculating the numerator:
\[ 42 + 195 = 237 \]
\[ 237 - 190 = 47 \]
\[ 47 + 186 = 233 \]
\[ 233 - 161 = 72 \]

Thus, we have:
\[ \frac{72}{210} \]

### Step 6: Simplify the Fraction
Now we simplify \(\frac{72}{210}\):
\[ \text{GCD of 72 and 210 is 6} \]
\[ \frac{72 \div 6}{210 \div 6} = \frac{12}{35} \]

### Final Result
The value of the expression is:
\[ \frac{12}{35} \]