Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is

Question

Accepted Answer

The problem involves calculating the speed of Anil based on the relative motion of two individuals traveling between two places. Here's the step-by-step solution:

**Key Information:**

- Distance between A and B = 45 km.
   - Anil and Sunil cross each other in **1 hour 30 minutes** (or **1.5 hours**).
   - Anil reaches B **1 hour 15 minutes** (or **1.25 hours**) after Sunil reaches A.

**Relative Speed:**

- Let Anil's speed be $$v_a$$ km/hr and Sunil's speed be $$v_s$$ km/hr.
   - Since they cross each other in **1.5 hours**, their relative speed is:
     $$
     v_a + v_s = \frac{	ext{Total Distance}}{	ext{Time Taken to Cross Each Other}} = \frac{45}{1.5} = 30 \, 	ext{km/hr}.
     $$

**Time Relation Between Anil and Sunil:**

- Let Sunil's time to travel from B to A be $$T$$ hours.
   - Anil's time to travel from A to B is $$T + 1.25$$ hours.

**Individual Speeds:**

- Sunil's speed:
     $$
     v_s = \frac{	ext{Distance}}{	ext{Time}} = \frac{45}{T}.
     $$
   - Anil's speed:
     $$
     v_a = \frac{	ext{Distance}}{	ext{Time}} = \frac{45}{T + 1.25}.
     $$

**Solve for $$T$$:**

Using the relative speed equation:
   $$
   v_a + v_s = 30
   $$
   Substitute values:
   $$
   \frac{45}{T + 1.25} + \frac{45}{T} = 30
   $$
   Multiply through by $$T(T + 1.25)$$ to eliminate the denominators:
   $$
   45T + 45(T + 1.25) = 30T(T + 1.25)
   $$
   Expand:
   $$
   45T + 45T + 56.25 = 30T^2 + 37.5T
   $$
   Rearrange terms:
   $$
   30T^2 - 52.5T - 56.25 = 0
   $$
   Simplify further by dividing through by $$7.5$$:
   $$
   4T^2 - 7T - 7.5 = 0
   $$

**Solve Quadratic Equation:**

Use the quadratic formula:
   $$
   T = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$
   Here, $$a = 4$$, $$b = -7$$, and $$c = -7.5$$:
   $$
   T = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(4)(-7.5)}}{2(4)}
   $$
   Simplify:
   $$
   T = \frac{7 \pm \sqrt{49 + 120}}{8} = \frac{7 \pm \sqrt{169}}{8} = \frac{7 \pm 13}{8}.
   $$
   Solve for $$T$$:
   - $$T = \frac{20}{8} = 2.5$$ (valid since time cannot be negative).

**Find Anil's Speed:**

Substitute $$T$$ into Anil's speed formula:
   $$
   v_a = \frac{45}{T + 1.25} = \frac{45}{2.5 + 1.25} = \frac{45}{3.75} = 12 \, 	ext{km/hr}.
   $$

### Final Answer:
The speed of Anil is **12 km/hr**.  
Correct option: **C**

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