A boat covers a distance of $400 ext{ km}$ downstream and $320 ext{ km}$ upstream in a total of $40$ hours. The downstream speed is $4 ext{ km/h}$ faster than the upstream speed. What is the time taken (in hours) by the boat to travel $720 ext{ km}$ downstream?

Question

A boat covers a distance of $400 	ext{ km}$ downstream and $320 	ext{ km}$ upstream in a total of $40$ hours. The downstream speed is $4 	ext{ km/h}$ faster than the upstream speed. What is the time taken (in hours) by the boat to travel $720 	ext{ km}$ downstream?

Accepted Answer

**Solution:**

1. Let the upstream speed be $x 	ext{ km/h}$. Then the downstream speed is $(x + 4) 	ext{ km/h}$.

2. The total time taken for downstream and upstream travel is given by:
   $$\frac{400}{x+4} + \frac{320}{x} = 40$$

3. Multiply both sides by $x(x+4)$ to clear denominators:
   $$400x + 320(x+4) = 40x(x+4)$$

4. Expand and simplify:
   $$400x + 320x + 1280 = 40x^2 + 160x$$
   $$720x + 1280 = 40x^2 + 160x$$

5. Rearrange terms:
   $$40x^2 + 160x - 720x - 1280 = 0$$
   $$40x^2 - 560x - 1280 = 0$$

6. Divide entire equation by 40:
   $$x^2 - 14x - 32 = 0$$

7. Solve quadratic equation:
   $$x = \frac{14 \pm \sqrt{14^2 + 4 	imes 32}}{2} = \frac{14 \pm \sqrt{196 + 128}}{2} = \frac{14 \pm \sqrt{324}}{2}$$
   $$x = \frac{14 \pm 18}{2}$$

8. Possible values:
   - $x = \frac{14 + 18}{2} = 16$ (valid speed)
   - $x = \frac{14 - 18}{2} = -2$ (not valid as speed cannot be negative)

9. So, upstream speed $= 16 	ext{ km/h}$ and downstream speed $= 16 + 4 = 20 	ext{ km/h}$.

10. Time taken to cover $720 	ext{ km}$ downstream:
    $$	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}} = \frac{720}{20} = 36 	ext{ hours}$$

**Answer:** $36$

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