Two cars travel from different locations at constant speeds. To meet each other after starting at the same time, they take 1.5 hours if they travel towards each other, but 10.5 hours if they travel in the same direction. If the speed of the slower car is 60 km/hr, then the distance traveled, in km, by the slower car when it meets the other car while traveling towards each other, is

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Accepted Answer

1. **Relative Speed When Traveling Towards Each Other:**  
   - When the cars are traveling towards each other, their relative speed is the sum of their individual speeds:  
     $$
     	ext{Relative Speed} = a + b
     $$  
   - Time taken to meet = 1.5 hours.  
   - Distance between the cars = $$ D = (a + b) 	imes 1.5 $$.

2. **Relative Speed When Traveling in the Same Direction:**  
   - When the cars are traveling in the same direction, their relative speed is the difference of their speeds:  
     $$
     	ext{Relative Speed} = a - b
     $$  
   - Time taken to meet = 10.5 hours.  
   - Distance between the cars = $$ D = (a - b) 	imes 10.5 $$.

3. **Relating Speeds and Solving for Ratio:**  
   - Since the distance between the cars is the same in both cases:  
     $$
     (a + b) 	imes 1.5 = (a - b) 	imes 10.5.
     $$  
   - Simplify:  
     $$
     a + b = 7(a - b).
     $$  
   - Expand and simplify further:  
     $$
     a + b = 7a - 7b \implies 8b = 6a \implies \frac{a}{b} = \frac{4}{3}.
     $$

4. **Substitute Known Value of Slower Car's Speed:**  
   - The slower car's speed $$ b = 60 $$ km/hr.  
   - Using $$ \frac{a}{b} = \frac{4}{3} $$, calculate faster car's speed:  
     $$
     a = \frac{4}{3} 	imes 60 = 80 	ext{ km/hr}.
     $$

5. **Distance Between Cars When Traveling Towards Each Other:**  
   - Relative speed when traveling towards each other:  
     $$
     a + b = 80 + 60 = 140 	ext{ km/hr}.
     $$  
   - Distance between cars:  
     $$
     D = (a + b) 	imes 1.5 = 140 	imes 1.5 = 210 	ext{ km}.
     $$

6. **Distance Traveled by Slower Car Before Meeting:**  
   - The slower car travels at a speed of $$ b = 60 $$ km/hr for 1.5 hours:  
     $$
     	ext{Distance Traveled} = b 	imes 	ext{time} = 60 	imes 1.5 = 90 	ext{ km}.
     $$

Thus, the distance traveled by the slower car is **90 km**, and the correct answer is **C**.

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