Consider the following representation of oxy-acid, `H_(n_(1))S_(2)O_(n_(2))`, (where S is central sulphur atom annd `n_(1) and n_(2)` are natural numbers.) if there are two possible oxy-acid of sulphur A and B contains ratio of `n_(2):n_(1)` are 2 and 4 respectively, then sum of oxidation state of 'S' atom in both oxy-acid will be:

Explanation

To solve the problem, we need to determine the oxidation states of sulfur in two oxyacids of sulfur, A and B, based on the given ratios of \( n_2:n_1 \) which are 2 and 4 respectively.

### Step-by-Step Solution:

1. **Identify the General Formula**:
The general formula for the oxyacid is \( H_{n_1}S_2O_{n_2} \).

2. **Determine the Oxyacids**:
Given the ratios:
- For oxyacid A, \( n_2:n_1 = 2 \): This means \( n_2 = 2 \) and \( n_1 = 1 \). Thus, the formula is \( H_2S_2O_4 \) (which is sulfuric acid).
- For oxyacid B, \( n_2:n_1 = 4 \): This means \( n_2 = 4 \) and \( n_1 = 2 \). Thus, the formula is \( H_2S_2O_8 \) (which is peroxy disulfuric acid).

3. **Calculate the Oxidation State for Oxyacid A (H2S2O4)**:
- Let the oxidation state of sulfur be \( x \).
- The equation based on the charge balance is:
\[ 2(+1) + 2x + 4(-2) = 0 \]
- Simplifying this:
\[ 2 + 2x - 8 = 0 \implies 2x - 6 = 0 \implies 2x = 6 \implies x = +3 \]
- Therefore, the oxidation state of sulfur in \( H_2S_2O_4 \) is +3.

4. **Calculate the Oxidation State for Oxyacid B (H2S2O8)**:
- Again, let the oxidation state of sulfur be \( y \).
- The equation based on the charge balance is:
\[ 2(+1) + 2y + 8(-2) = 0 \]
- Simplifying this:
\[ 2 + 2y - 16 = 0 \implies 2y - 14 = 0 \implies 2y = 14 \implies y = +7 \]
- Therefore, the oxidation state of sulfur in \( H_2S_2O_8 \) is +7.

5. **Sum of Oxidation States**:
- Now, we sum the oxidation states of sulfur from both oxyacids:
\[ +3 + +7 = +10 \]

### Final Answer:
The sum of the oxidation states of sulfur in both oxyacids A and B is **10**.