Find the downstream speed of the boat given the following statements:

Statement I: The difference between the downstream speed of the boat and the speed of the boat in still water is 12 km/hr.

Statement II: The ratio of the upstream speed of the boat to the speed of the boat in still water is 4:7.

Statement III: The total time taken by the boat to cover 280 km downstream and upstream is 24.5 hours.

Which of the following statements or combinations of statements are sufficient to find the downstream speed of the boat?

Question

Find the downstream speed of the boat given the following statements:

Statement I: The difference between the downstream speed of the boat and the speed of the boat in still water is 12 km/hr.

Statement II: The ratio of the upstream speed of the boat to the speed of the boat in still water is 4:7.

Statement III: The total time taken by the boat to cover 280 km downstream and upstream is 24.5 hours.

Which of the following statements or combinations of statements are sufficient to find the downstream speed of the boat?

Accepted Answer

**Solution:**

Let the speed of the boat in still water be *b* km/hr and the speed of the stream be *s* km/hr.

- Downstream speed = *b + s*
- Upstream speed = *b - s*

**Given Statements:**

1. Statement I: Downstream speed - speed in still water = 12
   $$ (b + s) - b = s = 12 	ext{ km/hr} $$
   So, the stream speed *s* is 12 km/hr.

2. Statement II: Ratio of upstream speed to speed in still water is 4:7
   $$ rac{b - s}{b} = rac{4}{7} $$
   Rearranged:
   $$ 7(b - s) = 4b \ 7b - 7s = 4b \ 3b = 7s \ b = rac{7s}{3} $$

3. Statement III: Total time for 280 km downstream and upstream is 24.5 hours
   $$ rac{280}{b + s} + rac{280}{b - s} = 24.5 $$

**Analysis:**

- Using Statement I alone, we know *s = 12* but *b* is unknown.
- Using Statement II alone, we have a relation between *b* and *s* but no numerical values.
- Using Statement III alone, we have one equation with two unknowns (*b* and *s*), insufficient.

**Combining Statements:**

- Statements I & III:
  From I, *s = 12*; substitute in III:
  $$ rac{280}{b + 12} + rac{280}{b - 12} = 24.5 $$
  This single equation can be solved for *b*.

- Statements II & III:
  From II, express *b* in terms of *s*:
  $$ b = rac{7s}{3} $$
  Substitute in III:
  $$ rac{280}{rac{7s}{3} + s} + rac{280}{rac{7s}{3} - s} = 24.5 $$
  This can be solved for *s*, then *b*.

- Statements I & II alone:
  From I, *s = 12*; from II, *b = rac{7 	imes 12}{3} = 28$ km/hr.
  Downstream speed = *b + s = 28 + 12 = 40$ km/hr.
  So, I & II together are sufficient.

- Statements I & II & III together obviously sufficient.

**Conclusion:**

- Either (I & III) or (II & III) together are sufficient to find downstream speed.
- I & II together also suffice.

However, since the options do not include I & II alone, the correct choice is:

**Answer:** Either statement I & III or statement II & III together

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