Find:
`1(10)/(21)+2(1)/(6)`

Options

A. `3(9)/(14)`
B. `3(8)/(14)`
C. `3(7)/(14)`
D. `3(6)/(14)`

Answer

To solve the problem \( 1\frac{10}{21} + 2\frac{1}{6} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers into improper fractions.

For \( 1\frac{10}{21} \):
- Multiply the whole number (1) by the denominator (21):
\[ 1 \times 21 = 21 \]
- Add the numerator (10) to this product:
\[ 21 + 10 = 31 \]
- So, \( 1\frac{10}{21} \) becomes:
\[ \frac{31}{21} \]

For \( 2\frac{1}{6} \):
- Multiply the whole number (2) by the denominator (6):
\[ 2 \times 6 = 12 \]
- Add the numerator (1) to this product:
\[ 12 + 1 = 13 \]
- So, \( 2\frac{1}{6} \) becomes:
\[ \frac{13}{6} \]

### Step 2: Add the Improper Fractions
Now we need to add \( \frac{31}{21} \) and \( \frac{13}{6} \). To do this, we need a common denominator.

### Step 3: Find the Least Common Multiple (LCM)
The denominators are 21 and 6. We need to find the LCM of these two numbers.

- The prime factorization of 21 is \( 3 \times 7 \).
- The prime factorization of 6 is \( 2 \times 3 \).

The LCM is found by taking the highest power of each prime:
- From 21: \( 3^1 \) and \( 7^1 \)
- From 6: \( 2^1 \)

Thus, the LCM is:
\[ 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42 \]

### Step 4: Convert Each Fraction to Have the Common Denominator
Now we convert each fraction to have a denominator of 42.

For \( \frac{31}{21} \):
- Multiply the numerator and denominator by 2:
\[ \frac{31 \times 2}{21 \times 2} = \frac{62}{42} \]

For \( \frac{13}{6} \):
- Multiply the numerator and denominator by 7:
\[ \frac{13 \times 7}{6 \times 7} = \frac{91}{42} \]

### Step 5: Add the Converted Fractions
Now we can add the two fractions:
\[ \frac{62}{42} + \frac{91}{42} = \frac{62 + 91}{42} = \frac{153}{42} \]

### Step 6: Simplify the Result (if possible)
Now we simplify \( \frac{153}{42} \). We can check if there are any common factors.

The GCD of 153 and 42 is 3. So we divide both the numerator and denominator by 3:
\[ \frac{153 \div 3}{42 \div 3} = \frac{51}{14} \]

### Final Answer
Thus, the final answer is:
\[ \frac{51}{14} \]