Simplify the following :
`2""3/14-3""5/6-2/5+2""1/2`.

Answer

To simplify the expression \( 2 \frac{3}{14} - 3 \frac{5}{6} - \frac{2}{5} + 2 \frac{1}{2} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \( 2 \frac{3}{14} \) to an improper fraction:
\[ 2 \frac{3}{14} = \frac{2 \times 14 + 3}{14} = \frac{28 + 3}{14} = \frac{31}{14} \]

2. Convert \( 3 \frac{5}{6} \) to an improper fraction:
\[ 3 \frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6} \]

3. Convert \( 2 \frac{1}{2} \) to an improper fraction:
\[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} \]

Now, the expression becomes:
\[ \frac{31}{14} - \frac{23}{6} - \frac{2}{5} + \frac{5}{2} \]

### Step 2: Find the Least Common Multiple (LCM) of the Denominators
The denominators are 14, 6, 5, and 2. We will find the LCM of these numbers.

- The prime factorization:
- \( 14 = 2 \times 7 \)
- \( 6 = 2 \times 3 \)
- \( 5 = 5 \)
- \( 2 = 2 \)

The LCM is found by taking the highest power of each prime:
- \( 2^1 \)
- \( 3^1 \)
- \( 5^1 \)
- \( 7^1 \)

Thus, the LCM is:
\[ LCM = 2^1 \times 3^1 \times 5^1 \times 7^1 = 2 \times 3 \times 5 \times 7 = 210 \]

### Step 3: Convert Each Fraction to Have the Same Denominator
Now we will convert each fraction to have a denominator of 210.

1. For \( \frac{31}{14} \):
\[ \frac{31}{14} = \frac{31 \times 15}{14 \times 15} = \frac{465}{210} \]

2. For \( \frac{23}{6} \):
\[ \frac{23}{6} = \frac{23 \times 35}{6 \times 35} = \frac{805}{210} \]

3. For \( \frac{2}{5} \):
\[ \frac{2}{5} = \frac{2 \times 42}{5 \times 42} = \frac{84}{210} \]

4. For \( \frac{5}{2} \):
\[ \frac{5}{2} = \frac{5 \times 105}{2 \times 105} = \frac{525}{210} \]

### Step 4: Rewrite the Expression
Now we can rewrite the expression:
\[ \frac{465}{210} - \frac{805}{210} - \frac{84}{210} + \frac{525}{210} \]

### Step 5: Combine the Fractions
Combine the fractions over the common denominator:
\[ \frac{465 - 805 - 84 + 525}{210} \]

Calculating the numerator:
\[ 465 - 805 = -340 \]
\[ -340 - 84 = -424 \]
\[ -424 + 525 = 101 \]

So we have:
\[ \frac{101}{210} \]

### Final Answer
The simplified form of the expression is:
\[ \frac{101}{210} \]