Find the difference between the age of Z five years from now and the age of X seven years ago.

Statement I: Age of Z eight years from now will be eight years less than the average age of X and Y at that time.

Statement II: Twelve years from now, the age of Y will be 52 years. The present age ratio of X to Y is 3:5.

Statement III: The age of X is 50% more than that of Z.

Question

Find the difference between the age of Z five years from now and the age of X seven years ago.

Statement I: Age of Z eight years from now will be eight years less than the average age of X and Y at that time.

Statement II: Twelve years from now, the age of Y will be 52 years. The present age ratio of X to Y is 3:5.

Statement III: The age of X is 50% more than that of Z.

Accepted Answer

**Solution:**

We need to find the difference between Z's age 5 years hence and X's age 7 years ago.

Let's analyze the statements:

1. **Statement I:** "Age of Z eight years hence will be eight years less than the average age of X and Y at that time."

Let present ages be $X$, $Y$, and $Z$.

Eight years hence:
   - Age of Z = $Z + 8$
   - Average age of X and Y = $\frac{(X + 8) + (Y + 8)}{2} = \frac{X + Y + 16}{2}$

According to statement I:
   $$Z + 8 = \frac{X + Y + 16}{2} - 8$$
   Simplify:
   $$Z + 8 = \frac{X + Y + 16 - 16}{2} = \frac{X + Y}{2}$$
   So,
   $$2Z + 16 = X + Y$$

This relates $X$, $Y$, and $Z$ but alone is insufficient to find the required difference.

2. **Statement II:** "Twelve years hence, Y will be 52 years. The present age ratio of X to Y is 3:5."

From this:
   $$Y + 12 = 52 \Rightarrow Y = 40$$

Given ratio:
   $$\frac{X}{Y} = \frac{3}{5} \Rightarrow X = \frac{3}{5} 	imes 40 = 24$$

Now, from statement II alone, we know $X$ and $Y$ but not $Z$, so we cannot find the required difference.

3. **Statement III:** "The age of X is 50% more than that of Z."

This means:
   $$X = Z + 0.5Z = 1.5Z$$

This relates $X$ and $Z$.

**Combining Statements:**

- From II and III:
  - $X = 24$
  - $X = 1.5Z \Rightarrow Z = \frac{24}{1.5} = 16$

- Now, we can find the required difference:
  - Age of Z five years hence = $16 + 5 = 21$
  - Age of X seven years ago = $24 - 7 = 17$
  - Difference = $21 - 17 = 4$

- From I and II:
  - From I: $2Z + 16 = X + Y$
  - Substitute $X=24$, $Y=40$ from II:
    $$2Z + 16 = 24 + 40 = 64 \Rightarrow 2Z = 48 \Rightarrow Z = 24$$
  - Then difference:
    - $Z + 5 = 24 + 5 = 29$
    - $X - 7 = 24 - 7 = 17$
    - Difference = $29 - 17 = 12$

- From I and III:
  - From III: $X = 1.5Z$
  - From I: $2Z + 16 = X + Y = 1.5Z + Y$
  - Rearranged: $2Z + 16 = 1.5Z + Y \Rightarrow Y = 0.5Z + 16$

Without knowing $Y$, we cannot find exact ages, so I and III alone are insufficient.

**Conclusion:**
- Statements II and III together are sufficient.
- Statements I and II together are sufficient.
- Statements I and III alone are not sufficient.

Hence, the correct choice is:

**Either statement I & II together or statement II & III together.**

**Answer:** Either statement I & II together or statement II & III together

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