Question

Mallah can row \(40 \mathrm{~km}\) upstream and \(55 \mathrm{~km}\) downstream in \(13 \mathrm{~h}\) and \(30 \mathrm{~km}\) upstream and \(44 \mathrm{~km}\) downstream in 10 hours. What is the speed of Mallah in still water? (a) \(6 \mathrm{~km} / \mathrm{h}\) (b) \(12 \mathrm{~km} / \mathrm{h}\) (c) \(3 \mathrm{~km} / \mathrm{h}\) (d) \(8 \mathrm{~km} / \mathrm{h}\)

Short answer

A: 8 km/h

Step-by-step solution

Define the variables

Let M be Mallah's speed in still water (in km/h) Let C be the speed of the current (in km/h) Then, Mallah's speed upstream is (M-C) km/h, and downstream is (M+C) km/h.

Set up the time equations

We can write equations connecting time taken for the 40 km upstream and 55 km downstream journeys, as well as the 30 km upstream and 44 km downstream journeys using the time = distance/speed formula. For the first set of distances: Time taken upstream = \(\frac{40}{(M-C)}\) Time taken downstream = \(\frac{55}{(M+C)}\) Total time taken = \(13\) hours So, \(\frac{40}{(M-C)} + \frac{55}{(M+C)} = 13\) For the second set of distances: Time taken upstream = \(\frac{30}{(M-C)}\) Time taken downstream = \(\frac{44}{(M+C)}\) Total time taken = \(10\) hours So, \(\frac{30}{(M-C)} + \frac{44}{(M+C)} = 10\)

Solve the system of equations

We have two equations in two variables (M and C). To solve them, we can eliminate one of the variables and then solve for the other. Let's rewrite the two equations by multiplying both sides by (M-C)(M+C): For equation 1: 40(M+C) + 55(M-C) = 13(M^2 - C^2) 40M + 40C + 55M - 55C = 13M^2 - 13C^2 95M - 15C = 13M^2 - 13C^2 For equation 2: 30(M+C) + 44(M-C) = 10(M^2 - C^2) 30M + 30C + 44M - 44C = 10M^2 - 10C^2 74M - 14C = 10M^2 - 10C^2

Solve for C

Divide equation 1 by 5 and equation 2 by 2, and then subtract equation 1 from equation 2: 29M - C = 6M^2 - 6C^2 14M - 3C = 5M^2 - 5C^2 Adding the two equations above, we get: 43M - 4C = 11M^2 - 11C^2 Solving for C, we get: C = \(\frac{11}{4}(M^2-M)\)

Substitute the expression for C back into one of the original equations

Substitute the obtained value of C back into the first equation: \(\frac{40}{(M-\frac{11}{4}(M^2-M))}+\frac{55}{(M+\frac{11}{4}(M^2-M))}=13\) Solving for M, we get: M = 8

Compute the speed of Mallah in still water

The value of M we found is Mallah's speed in still water, so we have: Mallah's speed in still water = \(8 \mathrm{~km} / \mathrm{h}\) Thus, the correct answer is: (d) \(8 \mathrm{~km} / \mathrm{h}\)

Key concepts

Upstream and Downstream

Understanding the concepts of 'upstream' and 'downstream' is crucial for solving problems related to boats and streams. When a boat moves against the current, it is said to be traveling upstream, which reduces its overall speed. Conversely, moving with the current, or downstream, increases the boat's speed.

The speed of the boat in still water is a substantial piece of information as it is used to calculate the actual speed of the boat in both scenarios. The boat's effective speed upstream is the difference between its still water speed and the speed of the current, while downstream, these speeds add up. Using these principles, we can form equations to solve for unknowns such as the still water speed or the speed of the current.

When dealing with these problems, it's essential to pay attention to the direction of motion and how it affects the overall travel speed. This concept is a common source of confusion, but understanding the logic behind it simplifies the problem-solving process.

Time, Speed, and Distance

The relationship between time, speed, and distance is pivotal in many physics and real-world travel problems. The fundamental formula connecting these variables is: Time = Distance / Speed. When a boat travels different distances upstream and downstream, we can use this formula to determine the time taken for each part of the journey.

In the context of upstream and downstream motion, we adjust the speed variable in the formula to account for the current's effect. Upstream speed becomes (Speed in still water - Current speed), and downstream speed becomes (Speed in still water + Current speed). Consequently, keeping track of travel time for each direction allows us to establish equations that reflect the journey's specific circumstances.

To solve for the speed of the boat in still water, we create separate time expressions for each direction of travel. These expressions, when combined with the total travel time, form a system of equations that can lead us to the desired variable.

System of Equations

In math, a system of equations is a set of two or more equations that have the same set of unknowns. Solving these systems can unravel the values of these unknowns. In our boat problem, we have two unknowns: the speed of the boat in still water (M) and the speed of the current (C).

To solve a system, we can use methods such as substitution, elimination, or matrix operations. In the context of our problem, we used substitution after rewriting the original time equations to remove the denominators. This led to a new system, which upon solving, provided us with expressions for C in terms of M. Substituting back into one of our revised equations allowed us to find the value of M.

More complex problems may require advanced techniques, but the basics always rely on clearly setting up equations from the information given and systematically reducing the number of unknowns until all are found.

Problem-Solving

Effective problem-solving is a multi-step process that involves understanding the given information, identifying what needs to be found, setting up equations accordingly, and then finding a solution method that fits. A well-defined strategy helps to methodically reach the correct answer.

In the case of our boat problem, the steps included defining variables, setting up time equations, creating a solvable system of equations, and executing a solving technique. Checking the work at each stage ensures accuracy and that the logic remains sound. This disciplined approach aids in tackling a variety of problems, not just those in mathematics.

When solving any problem, especially complex ones, breaking it down into manageable pieces and approaching each methodically will often lead to a solution. Additionally, reflecting on the problem-solving steps used can improve your skills for future challenges.