Which of the following statement/s is or are required to find the ratio of age of A after 6 years and age of B two years ago?

Statement 1: The sum of present ages of A and B is 78 years.

Statement 2: B is 34 years older than A and 12 years older than C.

Statement 3: Ratio of present ages of A and C is 1:2.

Question

Which of the following statement/s is or are required to find the ratio of age of A after 6 years and age of B two years ago?

Statement 1: The sum of present ages of A and B is 78 years.

Statement 2: B is 34 years older than A and 12 years older than C.

Statement 3: Ratio of present ages of A and C is 1:2.

Accepted Answer

**Solution:**

We need to find the ratio of age of A after 6 years to age of B two years ago, i.e.,

$$\frac{A + 6}{B - 2}$$

where *A* and *B* represent the present ages of A and B respectively.

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**Step 1: Analyze Statement 1**
- Statement 1 says: $A + B = 78$
- This gives a relation between *A* and *B* but not their individual values.
- Alone, this is insufficient to find the required ratio.

**Step 2: Analyze Statement 2**
- Statement 2 says: $B = A + 34$ and $B = C + 12$
- This relates *B* to *A* and *C*.
- Using $B = A + 34$, we can substitute in the ratio:
  $$\frac{A + 6}{(A + 34) - 2} = \frac{A + 6}{A + 32}$$
- But we still don't know *A*.
- Alone, this is insufficient.

**Step 3: Analyze Statement 3**
- Statement 3 says: $\frac{A}{C} = \frac{1}{2}$ or $C = 2A$
- Alone, this does not relate *B* and *A* directly.
- Insufficient alone.

**Step 4: Combine Statements 1 and 2**
- From Statement 2: $B = A + 34$
- Substitute in Statement 1: $A + (A + 34) = 78 \Rightarrow 2A + 34 = 78 \Rightarrow 2A = 44 \Rightarrow A = 22$
- Then $B = 22 + 34 = 56$
- Now find the ratio:
  $$\frac{A + 6}{B - 2} = \frac{22 + 6}{56 - 2} = \frac{28}{54} = \frac{14}{27}$$
- So Statements 1 and 2 together are sufficient.

**Step 5: Combine Statements 2 and 3**
- From Statement 3: $C = 2A$
- From Statement 2: $B = C + 12 = 2A + 12$ and also $B = A + 34$
- Equate: $A + 34 = 2A + 12 \Rightarrow 34 - 12 = 2A - A \Rightarrow 22 = A$
- Then $B = 22 + 34 = 56$
- Ratio:
  $$\frac{A + 6}{B - 2} = \frac{22 + 6}{56 - 2} = \frac{28}{54} = \frac{14}{27}$$
- So Statements 2 and 3 together are sufficient.

**Step 6: Combine Statements 1 and 3**
- From Statement 3: $C = 2A$
- Statement 1: $A + B = 78$
- No direct relation between *B* and *A* here, so cannot find *B* or the ratio.
- Insufficient.

**Step 7: All statements together**
- Since 1 & 2 or 2 & 3 are sufficient, all together are also sufficient but not necessary.

**Answer:** (a) either 1 & 2 together or 2 & 3 together

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