Quantity I : B' s present age is `60%` more than A' s present age and ratio of present age of B to that of C is 5 : 2 D is 8 years younger than B and D' present age is twice of that of C . Find average of present age of A, B C & D (in years )
Quantity II : Present age of R is equal to average of present age of P & Q 4 years hence, age of P is twice of age of Q at that time . If R is 15 - years younger tan P then find age of younger person among p , Q & r

Options

A. Quantity I ` lt ` Quantity II
B. Quantity I ` le ` Quantity II
C. Quantity I ` gt ` Quantity II
D. Quantity I ` ge ` Quantity II

Answer

To solve the problem, we need to break it down into two parts: Quantity I and Quantity II.

### Quantity I
1. **Let A's present age be \( x \) years.**
- Since B's present age is 60% more than A's, we can express B's age as:
\[ B = x + 0.6x = 1.6x \]

2. **The ratio of B's age to C's age is given as 5:2.**
- This means:
\[ \frac{B}{C} = \frac{5}{2} \]
- Substituting B's age:
\[ \frac{1.6x}{C} = \frac{5}{2} \]
- Cross-multiplying gives:
\[ 2 \cdot 1.6x = 5C \implies C = \frac{3.2x}{5} = 0.64x \]

3. **D is 8 years younger than B.**
- Therefore, D's age can be expressed as:
\[ D = B - 8 = 1.6x - 8 \]

4. **D's present age is also twice that of C.**
- Thus, we can set up the equation:
\[ D = 2C \]
- Substituting for D and C:
\[ 1.6x - 8 = 2(0.64x) \]
- Simplifying gives:
\[ 1.6x - 8 = 1.28x \]
- Rearranging:
\[ 1.6x - 1.28x = 8 \implies 0.32x = 8 \implies x = \frac{8}{0.32} = 25 \]

5. **Now we can find the ages of A, B, C, and D.**
- A's age:
\[ A = x = 25 \text{ years} \]
- B's age:
\[ B = 1.6x = 1.6 \times 25 = 40 \text{ years} \]
- C's age:
\[ C = 0.64x = 0.64 \times 25 = 16 \text{ years} \]
- D's age:
\[ D = 1.6x - 8 = 40 - 8 = 32 \text{ years} \]

6. **Calculate the average age of A, B, C, and D.**
- The average age is given by:
\[ \text{Average} = \frac{A + B + C + D}{4} = \frac{25 + 40 + 16 + 32}{4} = \frac{113}{4} = 28.25 \text{ years} \]

### Quantity II
1. **Let P's present age be \( p \) years and Q's present age be \( q \) years.**
- The average age of P and Q is given by:
\[ R = \frac{p + q}{2} \]

2. **Four years hence, P's age will be twice Q's age.**
- This can be expressed as:
\[ p + 4 = 2(q + 4) \]
- Simplifying gives:
\[ p + 4 = 2q + 8 \implies p = 2q + 4 \]

3. **R is 15 years younger than P.**
- Therefore:
\[ R = p - 15 \]

4. **Equating the two expressions for R:**
\[ \frac{p + q}{2} = p - 15 \]
- Cross-multiplying gives:
\[ p + q = 2(p - 15) \implies p + q = 2p - 30 \implies q = p - 30 \]

5. **Substituting \( q \) in terms of \( p \) into the equation \( p = 2q + 4 \):**
\[ p = 2(p - 30) + 4 \implies p = 2p - 60 + 4 \implies p - 2p = -56 \implies -p = -56 \implies p = 56 \]
- Then substituting back to find \( q \):
\[ q = 56 - 30 = 26 \]
- Finding R:
\[ R = 56 - 15 = 41 \]

6. **Identify the younger person among P, Q, and R:**
- Ages are:
\[ P = 56, \quad Q = 26, \quad R = 41 \]
- The youngest person is Q, who is 26 years old.

### Final Results
- **Quantity I:** Average age of A, B, C, and D is **28.25 years**.
- **Quantity II:** The age of the younger person among P, Q, and R is **26 years** (Q).