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CAT 2025Slot 2QAQues & Sol

ArithmeticHard

Question

Rita and Sneha can row a boat at 5 km/h and 6 km/h in still water, respectively. In a river flowing with a constant velocity, Sneha takes 48 minutes more to row 14 km upstream than to row the same distance downstream. If Rita starts from a certain location in the river, and returns downstream to the same location, taking a total of 100 minutes, then the total distance, in km, Rita will cover is

Solution

1. Concept Used

  • Topic: Time, Speed and Distance — Boats and Streams
  • Formula: $$ \text{Time} = \frac{\text{Distance}}{\text{Speed}}, \quad \text{Upstream Speed} = u - v, \quad \text{Downstream Speed} = u + v $$

2. Calculation

Step 1: Find the river speed using Sneha's data.

Let the river speed be ( v ) km/h. Sneha's still-water speed is 6 km/h, so her upstream speed is ( (6 - v) ) km/h and downstream speed is ( (6 + v) ) km/h.

Sneha takes 48 minutes ( = $\frac{48}{60} = 0.8 $) hours more to row 14 km upstream than downstream. Setting up the equation:

$$ \frac{14}{6 - v} - \frac{14}{6 + v} = 0.8 $$

Combining the fractions on the left:

$$ \frac{14(6 + v) - 14(6 - v)}{(6 - v)(6 + v)} = 0.8 $$

$$ \frac{14 \cdot 2v}{36 - v^2} = 0.8 $$

$$ \frac{28v}{36 - v^2} = 0.8 $$

Cross-multiplying: $$ 28v = 0.8(36 - v^2) = 28.8 - 0.8v^2 $$

Rearranging: $$ 0.8v^2 + 28v - 28.8 = 0 $$

Multiplying through by 5: $$ 4v^2 + 140v - 144 = 0 $$

Dividing by 4: $$ v^2 + 35v - 36 = 0 $$

Factoring: $$ (v + 36)(v - 1) = 0 $$

Since speed must be positive, $$ v = 1 \text{ km/h} $$

Step 2: Find the total distance Rita covers.

Rita's still-water speed is 5 km/h and the river speed is 1 km/h.

$$ \text{Rita's upstream speed} = 5 - 1 = 4 \text{ km/h} $$ $$ \text{Rita's downstream speed} = 5 + 1 = 6 \text{ km/h} $$

Let Rita row ( d ) km upstream and return ( d ) km downstream. Total time = 100 minutes $( = \frac{100}{60} = \frac{5}{3} )$ hours.

$$ \frac{d}{4} + \frac{d}{6} = \frac{5}{3} $$

$$ d \left(\frac{1}{4} + \frac{1}{6}\right) = \frac{5}{3} $$

$$ d \cdot \frac{3 + 2}{12} = \frac{5}{3} $$

$$ d \cdot \frac{5}{12} = \frac{5}{3} $$

$$ d = \frac{5}{3} \times \frac{12}{5} = 4 \text{ km} $$

Total distance covered by Rita $$ = 2d = 2 \times 4 = 8 \text{ km} $$

3. Solution

Answer = 8 ✅

The total distance Rita covers is 8 km.