P and Q compete in a 400 m race. P runs at a speed of $12 ext{ km/hr}$. P gives Q a head start of $20 ext{ m}$ and still wins the race by $13$ seconds. What is the speed of Q? (Round your answer to two decimal places.)

Question

P and Q compete in a 400 m race. P runs at a speed of $12 	ext{ km/hr}$. P gives Q a head start of $20 	ext{ m}$ and still wins the race by $13$ seconds. What is the speed of Q? (Round your answer to two decimal places.)

Accepted Answer

**Solution:**

1. Convert P's speed from km/hr to m/s:
   
   $$12 	imes \frac{5}{18} = \frac{60}{18} = \frac{10}{3} 	ext{ m/s}$$

2. Calculate the time taken by P to finish the race of 400 m:
   
   $$	ext{Time}_P = \frac{400}{\frac{10}{3}} = 400 	imes \frac{3}{10} = 120 	ext{ seconds}$$

3. Since P beats Q by 13 seconds, Q's time to finish his distance (380 m) is:
   
   $$	ext{Time}_Q = 120 + 13 = 133 	ext{ seconds}$$

4. Calculate Q's speed in m/s:
   
   $$	ext{Speed}_Q = \frac{380}{133} \approx 2.8571 	ext{ m/s}$$

5. Convert Q's speed back to km/hr:
   
   $$2.8571 	imes \frac{18}{5} = 10.2857 \approx 10.29 	ext{ km/hr}$$

**Answer:** $10.29 	ext{ km/hr}$

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