A question is given, followed by three statements labelled I, II and III. Identify which of the statements is/are sufficient to answer the question.

Question:

What is the value of (a - c)?

Statements:

I. (a + b) : c = 8 : 3 and a + b + c = 2772

II. a : (a + c) 2 : 5

III. a : (b + c) = 34 : 43

Correct option is D

Statement I:

Given:

• (a + b) : c = 8 : 3 → So, a + b = 8k, c = 3k
• Also, a + b + c = 2772

Substitute:

• a + b + c = 8k + 3k = 11k = 2772
  => k = 252
  So:
• a + b = 8k = 2016
• c = 3k = 756

We now know:
a + b = 2016
c = 756

But we still don't know a or b individually, so we can't get a − c directly.

So, Statement I alone is NOT sufficient.

Statement II:

Given:

• a : (a + c) = 2 : 5
  => a / (a + c) = 2 / 5
  Cross-multiplying: 5a = 2(a + c) => 5a = 2a + 2c => 3a = 2c => a = (2/3)c

We now have a in terms of c:

• a = (2/3)c => a − c = (2/3)c − c = (−1/3)c

But we don’t know c’s value, so Statement II alone is NOT sufficient.

Statement III:

Given:

• a : (b + c) = 34 : 43
  => a / (b + c) = 34 / 43
  Again, this gives a in terms of b + c:
• a = (34/43)(b + c)

But this is not enough to get either a or c directly.
So Statement III alone is NOT sufficient.

Check combinations:

I and II:

From I:

• a + b = 2016
• c = 756

From II:

• a = (2/3)c = (2/3) × 756 = 504
  Then b = 2016 − a = 2016 − 504 = 1512
  We know a = 504 and c = 756
  => a − c = 504 − 756 = −252

So I and II together are sufficient.

I and III:

From I:

• a + b = 2016
• c = 756
  Then b = 2016 − a

From III:

• a / (b + c) = 34 / 43
  Substitute b = 2016 − a and c = 756

Then:

​$$\frac{a}{(2016 - a + 756)} = \frac{34}{43} \Rightarrow \frac{a}{2772 - a} = \frac{34}{43}$$

Cross-multiplying: 43a = 34(2772 − a)
=> 43a = 94248 − 34a
=> 77a = 94248
=> a = 1224
Then c = 756 => a − c = 1224 − 756 = 468

So I and III together are also sufficient

Final Answer: (D) I and II or I and III​​