Question

\(A\) boat moves downstream at 1 km in 5 minutes as upstream at 1 km in 12 minutes. What is the speed of current? (b) \(3.5 \mathrm{~km} / \mathrm{h}\) (a) \(4.5 \mathrm{~km} / \mathrm{h}\) (d) \(2.5 \mathrm{~km} / \mathrm{h}\) (c) \(2 \mathrm{~km} / \mathrm{h}\)

Short answer

Answer: The speed of the current is 3.5 km/h.

Step-by-step solution

Write down the given information

The boat moves 1 km downstream in 5 minutes, and 1 km upstream in 12 minutes. We need to find the speed of the current.

Convert the given time in minutes to hours

To find the speed in km/h, we need to convert the given time from minutes to hours. We know that 1 hour = 60 minutes. So, we can convert 5 minutes and 12 minutes to hours as follows: 5 minutes = 5/60 hours = 1/12 hours 12 minutes = 12/60 hours = 1/5 hours

Set up equations for the speed of the boat downstream and upstream

Let's denote the speed of the boat in still water as "b" (in km/h) and the speed of the current as "c" (in km/h). When the boat moves downstream, its effective speed is (b + c) km/h. When it moves upstream, its effective speed is (b - c) km/h. We can now set up the equations for speeds downstream and upstream as follows: For downstream: (b + c) * (1/12) = 1 (1) b + c = 12 For upstream: (b - c) * (1/5) = 1 (2) b - c = 5

Solve the equations for 'c'

We can find the speed of the current by solving equations (1) and (2). We will add equation (1) and equation (2) to eliminate "b": (1) b + c = 12 (2) b - c = 5 ---------------- 2b = 17 Now, we can find "b" by dividing by 2: b = 17/2 = 8.5 km/h Now, substitute the value of 'b' in equation (1): 8.5 + c = 12 c = 12 - 8.5 = 3.5 km/h The speed of the current is 3.5 km/h, which is the answer (b).

Key concepts

Boat and Stream Problems

Understanding boat and stream problems is critical in mastering quantitative aptitude exercises. Such problems involve scenarios where a boat is moving on a body of water, affected by the current of the stream. The boat can move downstream (with the current) or upstream (against the current), both of which impact the effective speed of the boat.

The key to solving these problems is to characterize the different speeds involved: the speed of the boat in still water and the speed of the current. These can be combined or subtracted to find the boat's effective speed downstream or upstream. In the given exercise, we calculate these effective speeds based on the distance covered and the time taken. A deep understanding of these concepts enables students to set up and solve equations that relate to real-world situations involving motion in water.

Speed Time and Distance

Core to many quantitative problems, the relationship between speed, time, and distance is quintessential. The basic formula connecting these three elements is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
To apply this to boat and stream problems, one must remember to convert all units to match, often converting minutes to hours to find the speed in km/h. From the case study, we initially find the boat's speed by utilizing the time taken to travel a fixed distance downstream and upstream. By expressing these speeds in terms of the boat’s speed in still water plus or minus the speed of the current, we can set up equations that help us solve for the unknown variable, usually the current speed or the boat's speed in still water.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numbers and solve mathematical issues with speed and accuracy. It includes various topics such as arithmetic, algebra, geometry, and data interpretation. Boat and stream problems, being a part of this larger field, test a person’s capability to apply mathematical concepts to real-world scenarios. To excel in these problems, one should focus on practicing different types of questions and familiarizing oneself with quick methods to solve equations.

The exercise provided is an excellent example of how quantitative aptitude extends beyond rote memorization of formulas into the territory of critical thinking and problem-solving. It showcases how multiple concepts can intertwine — in this case, understanding the effects of current on speed while employing algebraic methods to find a solution.