Question

A motorboat whose speed in still water is 24 km/hr, takes 1 hour more to go 32 km upstream than to return downstream to the same spot.

Based on this information, answer the following questions.

1. What is the speed of the motorboat in going upstream, if speed of the stream is $$x$$ km/hr?

(a) $$(x - 24)$$ km/hr

(b) $$(24 - x)$$ km/hr

(c) $$(x + 24)$$ km/hr

(d) none of these

2. The quadratic equation which represents the given information is :

(a) $$x^2 - 64x - 576 = 0$$

(b) $$x^2 + 64x + 576 = 0$$

(c) $$x^2 + 64x - 576 = 0$$

(d) $$x^2 - 64x + 576 = 0$$

3. Speed of the motorboat in going downstream is :

(a) 16 km/hr

(b) 8 km/hr

(c) 28 km/hr

(d) 32 km/hr

4. Time taken by the motorboat to go 272 km downstream is :

(a) $$8\frac{1}{2}$$ hours

(b) 17 hours

(c) $$12\frac{1}{2}$$ hours

(d) $$6\frac{1}{2}$$ hours

5. Time taken by the motorboat to go 80 km upstream and then to return back to the same spot is :

(a) $$5\frac{1}{2}$$ hours

(b) $$6\frac{1}{2}$$ hours

(c) $$7\frac{1}{2}$$ hours

(d) $$8\frac{1}{2}$$ hours

Answer 1

1. The correct option is (b).

Reasoning:

Speed of the motorboat in still water = 24 km/hr

Speed of the stream = x km/hr

When going upstream, the boat moves against the stream, so the effective speed is the speed of the boat minus the speed of the stream: \[\text{Speed upstream} = (24 - x)\text{ km/hr}\]

2. The correct option is (c).

Reasoning:

Upstream speed = (24 – x) km/hr

Downstream speed = (24 + x) km/hr

Distance = 32 km

Difference in time = 1 hour

\[\frac{32}{24 - x} - \frac{32}{24 + x} = 1\] 

\[32 \left[\frac{(24 + x) - (24 - x)}{(24 - x)(24 + x)}\right] = 1\] 

\[32 \left[\frac{2x}{576 - x^2}\right] = 1\] 

\[64x = 576 - x^2\] 

\[x^2 + 64x - 576 = 0\]

3. The correct option is (d).

Reasoning: Solving the quadratic equation:

\[x^2 + 64x - 576 = 0\] 

\[x^2 + 72x - 8x - 576 = 0\] 

\[x(x + 72) - 8(x + 72) = 0\] 

\[(x + 72)(x - 8) = 0\] 

\[x = 8 \quad \text{or} \quad x = -72\]

Since stream speed cannot be negative, \[x = 8\text{ km/hr}\].

Now, speed downstream is \[\text{Speed downstream} = 24 + x = 24 + 8 = 32\text{ km/hr}\]

4. The correct option is (a).

Reasoning:

Distance = 272 km

Speed downstream = 32 km/hr

\[\text{Time taken} = \frac{\text{Distance}}{\text{Speed downstream}}\]

\[= \frac{272}{32}\]

\[= \frac{17}{2}\]

\[= 8\frac{1}{2}\text{ hours}\]

5. The correct option is (c).

Reasoning:

Speed upstream = \[24 - 8 = 16\text{ km/hr}\]

Speed downstream = \[24 + 8 = 32\text{ km/hr}\]

Distance each way = 80 km

\[\text{Time upstream} = \frac{80}{16} = 5\text{ hours}\]

\[\text{Time downstream} = \frac{80}{32} = 2.5\text{ hours} = 2\frac{1}{2}\text{ hours}\]

\[\text{Total time} = 5 + 2\frac{1}{2} = 7\frac{1}{2}\text{ hours}\]