Question

The speed of a boat in still water is $$x$$ km/hr and the speed of the stream is 3 km/hr. 

1. Write the speed of the boat upstream, in terms of $$x$$. 
2. Write the speed of the boat downstream, in terms of $$x$$. 
3. If the boat goes 15 km upstream and 22 km downstream in 5 hours, write an equation in $$x$$ to represent the statement. 
4. Solve the equation to evaluate $$x$$.

Answer 1

i. Speed upstream $$= (x - 3)\text{ km/hr}$$. 

ii. Speed downstream $$= (x + 3)\text{ km/hr}$$. 

iii. Time taken upstream $$= \frac{15}{x - 3}\text{ hours}$$ and downstream $$= \frac{22}{x + 3}\text{ hours}$$. 

Total time equation: $$\frac{15}{x - 3} + \frac{22}{x + 3} = 5$$ 

iv. Solving the equation: $$\frac{15(x + 3) + 22(x - 3)}{(x - 3)(x + 3)} = 5$$ 

$$\frac{15x + 45 + 22x - 66}{x^2 - 9} = 5$$

$$\frac{37x - 21}{x^2 - 9} = 5$$

$$37x - 21 = 5x^2 - 45$$

$$5x^2 - 37x - 24 = 0$$ 

Factoring: $$5x^2 - 40x + 3x - 24 = 0$$

$$5x(x - 8) + 3(x - 8) = 0$$

$$(x - 8)(5x + 3) = 0$$

$$x = 8 \quad \text{or} \quad x = -\frac{3}{5}$$ 

Since speed cannot be negative, reject $$x = -\frac{3}{5}$$.

Hence, $$x = 8$$.