A question is given, followed by three statements labelled I, II and III. Identify which of the statements is/are sufficient to answer the question.

Question:

How much time (in hours) will a boat take to go 25 km downstream?

Statements:

I. The time taken by the boat to go a certain distance downstream is of the time taken by it to go the same distance in still water. $$\frac{7}{10}$$​ 

II. The boat takes 6 hours to go 49 km downstream and 14 km upstream.

III. The ratio of the speed of the boat in still water and the speed of the current is 7 : 3.

Correct option is D

Given:

​Question:

How much time (in hours) will a boat take to go 25 km downstream?

Statements:

I. The time taken by the boat to go a certain distance downstream is 7/10​ of the time taken by it to go the same distance in still water.
II. The boat takes 6 hours to go 49 km downstream and 14 km upstream.
III. The ratio of the speed of the boat in still water and the speed of the current is 7 : 3.

Solution:

​$$\textbf{Using Statements I and II:} \\\text{Let speed of boat in still water} = b,\ \text{speed of current} = c \\\text{So, downstream speed} = b + c,\quad \text{still water speed} = b \\\text{From Statement I:} \\\frac{1}{b + c} = \frac{7}{10b} \Rightarrow 10b = 7(b + c) \\\ \\\Rightarrow 10b = 7b + 7c \Rightarrow 3b = 7c \Rightarrow \frac{b}{c} = \frac{7}{3} \\\text{Let } b = 7x,\ c = 3x \Rightarrow \text{Downstream speed} = b + c = 10x,\ \text{Upstream speed} = b - c = 4x\\[10pt]\text{From Statement II:} \\\frac{49}{10x} + \frac{14}{4x} = 6 \\\ \\\Rightarrow \frac{(49 \cdot 2 + 14 \cdot 5)}{20x} = \frac{98 + 70}{20x} = \frac{168}{20x} = 6 \\\ \\\Rightarrow 168 = 120x \Rightarrow x = \frac{168}{120} = 1.4\\[10pt]\text{Now, downstream speed} = 10x = 10 \cdot 1.4 = 14 \text{ km/h} \\\ \\\text{Time to travel 25 km downstream} = \frac{25}{14} = 1.7857 \text{ hours} \approx 1 \text{ hr } 47 \text{ min}\\[10pt]$$

Now Using Statement II and III Together
Statement III:
Boat : Current = 7 : 3
→ Let boat speed = 7x, current = 3x
→ Downstream = 10x, Upstream = 4x

This is the same data we deduced from I and III, and Statement II gives us:

​$$\frac{49}{10x} + \frac{14}{4x} = 6 \Rightarrow x = 1.4\\[10pt]\text{So again,} \quad \textbf{Downstream speed} = 10x = 14\ \text{km/h} \Rightarrow \text{Time for 25 km} = \frac{25}{14} \approx 1.785\ \text{hours}$$​​

Combine Statements II and III:

• Statement II gives time and distance total.
• Statement III gives the ratio of speeds, so we can calculate downstream and upstream speeds as a ratio of a common variable.
• With these, we can form equations and calculate downstream speed. Sufficient

I and III:

Without any distance or time values, just the ratio and proportion are not enough.

Final Answer: (D) I and II or II and III

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