Analyzing Boat Speed and Water Current This problem involves understanding the motion of a boat in water, specifically how its speed is affected by the water current when moving upstream and downstream. We are given the time taken to cover a certain distance in both directions and need to find the ratio of the boat's speed in still water to the speed of the water current. Defining Speeds and Distances Let's define the key speeds involved: Let v_b be the speed of the boat in still water. Let v_c be the speed of the water current. When the boat moves upstream, it is going against the water current, so the effective speed is reduced. When the boat moves downstream, it is going with the water current, so the effective speed is increased. Speed upstream = v_b - v_c Speed downstream = v_b + v_c Let D be the distance covered in one direction (from starting point to the destination and back). Converting Time Units The time taken for the upstream journey is given as 8 hours 48 minutes. We need to convert this time entirely into hours. 8 hours 48 minutes = 8 hours + 48 minutes To convert minutes to hours, divide by 60: 48 minutes = (48)/(60) hours = (4)/(5) hours = 0.8 hours. So, the upstream time is 8 + 0.8 = 8.8 hours. The time taken for the downstream journey is given as 4 hours. Setting Up Equations with Distance, Speed, and Time We know the formula: Distance = Speed × Time. Using this formula, we can set up equations for both the upstream and downstream journeys: For the upstream journey: D = (v_b - v_c) × 8.8 For the downstream journey: D = (v_b + v_c) × 4 Since the distance D is the same for both journeys, we can equate the right-hand sides of the two equations: (v_b - v_c) × 8.8 = (v_b + v_c) × 4 Solving for the Ratio of Speeds Now, we need to solve this equation to find the ratio v_b : v_c. First, distribute the numbers on both sides of the equation: 8.8 v_b - 8.8 v_c = 4 v_b + 4 v_c Next, gather the terms involving v_b on one side and the terms involving v_c on the other side. Let's move 4 v_b to the left side and -8.8 v_c to the right side. 8.8 v_b - 4 v_b = 4 v_c + 8.8 v_c Perform the subtraction on the left and the addition on the right: 4.8 v_b = 12.8 v_c To find the ratio v_b : v_c, we can divide both sides by v_c and by 4.8: (v_b)/(v_c) = (12.8)/(4.8) To simplify the ratio (12.8)/(4.8), we can remove the decimal points by multiplying the numerator and the denominator by 10: (v_b)/(v_c) = (128)/(48) Now, simplify the fraction (128)/(48) by dividing the numerator and denominator by their greatest common divisor. We can see that both numbers are divisible by 16 (128 = 16 × 8, 48 = 16 × 3). (v_b)/(v_c) = (128 ÷ 16)/(48 ÷ 16) = (8)/(3) So, the ratio of the speed of the boat in still water (v_b) to the speed of the water current (v_c) is 8 : 3. Conclusion on Boat Speed Ratio The calculated ratio of the speed of the boat in still water to that of the water current is 8 : 3. This means that for every 8 units of speed of the boat in still water, the speed of the current is 3 units. Parameter Value Upstream Time 8 hours 48 minutes = 8.8 hours Downstream Time 4 hours Speed Upstream v_b - v_c Speed Downstream v_b + v_c Equation (v_b - v_c) × 8.8 = (v_b + v_c) × 4 Ratio v_b : v_c 8 : 3 Revision Table: Boat and Stream Concepts Concept Explanation Formula Speed in Still Water The speed of the boat without any influence from the current. v_b Speed of Current The speed at which the water is flowing. v_c Upstream Speed The effective speed of the boat when moving against the current. v_{upstream} = v_b - v_c Downstream Speed The effective speed of the boat when moving with the current. v_{downstream} = v_b + v_c Distance, Speed, Time Relation Relates distance covered, speed, and time taken. Distance = Speed × Time Additional Information: Solving Boat and Stream Problems Boat and stream problems are common in time, speed, and distance calculations. They typically involve finding the speed of the boat, the speed of the current, the distance, or the time taken for journeys upstream or downstream. The core idea is to correctly calculate the effective speed by adding or subtracting the speed of the current from the boat's speed in still water. Key points to remember: Always ensure that units of speed, time, and distance are consistent. Convert minutes to hours or vice versa if necessary. The distance covered upstream and downstream is often the same in return journey problems, allowing you to equate the distance formulas. If you know the upstream speed (S_u) and downstream speed (S_d), you can directly find the speed of the boat and current: Speed of boat in still water v_b = (S_d + S_u)/(2) Speed of current v_c = (S_d - S_u)/(2) In this problem, we found the ratio by solving the equation 8.8(v_b - v_c) = 4(v_b + v_c). We could also have found the speeds relative to each other. Let S_u = v_b - v_c and S_d = v_b + v_c. We have D = S_u × 8.8 and D = S_d × 4. So, S_u × 8.8 = S_d × 4, which means (S_d)/(S_u) = (8.8)/(4) = (88)/(40) = (11)/(5). S_d : S_u = 11 : 5. So, v_b + v_c is proportional to 11, and v_b - v_c is proportional to 5. Using the formulas: v_b \propto (11+5)/(2) = (16)/(2) = 8 v_c \propto (11-5)/(2) = (6)/(2) = 3 Thus, v_b : v_c = 8 : 3. This confirms our previous calculation.