Solving Rowing Speed Problems: Upstream and Downstream

This problem involves the concepts of upstream and downstream motion, which are common in quantitative aptitude questions. When a boat travels upstream, its speed is reduced by the speed of the current. When it travels downstream, its speed is increased by the speed of the current.

Understanding Upstream and Downstream Speeds

• Let the speed of the boat in still water be \(b\) km/h.
• Let the speed of the current be \(c\) km/h.
• Speed of the boat upstream = \((b - c)\) km/h (Boat speed minus current speed).
• Speed of the boat downstream = \((b + c)\) km/h (Boat speed plus current speed).

The fundamental formula relating distance, speed, and time is:

\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

Setting Up Equations from the Given Information

We are given two scenarios:

1. Dharmendra rows 80 km upstream and 110 km downstream in a total of 13 hours.
2. He also rows 60 km upstream and 88 km downstream in a total of 10 hours.

Using the time formula for each part of the journey (upstream and downstream) and summing them up for each scenario, we get the following equations:

Scenario 1:

Time upstream + Time downstream = Total time

\(\frac{80}{b - c} + \frac{110}{b + c} = 13\) (Equation 1)

Scenario 2:

Time upstream + Time downstream = Total time

\(\frac{60}{b - c} + \frac{88}{b + c} = 10\) (Equation 2)

These are two linear equations in terms of \(\frac{1}{b - c}\) and \(\frac{1}{b + c}\). To make solving easier, let's substitute variables:

Let \(u = \frac{1}{b - c}\) and \(d = \frac{1}{b + c}\).

The equations become:

• \(80u + 110d = 13\) (Equation 3)
• \(60u + 88d = 10\) (Equation 4)

Solving the System of Linear Equations

We can solve this system using elimination. Let's multiply Equation 3 by a factor and Equation 4 by another factor so that the coefficients of \(u\) become equal.

• Multiply Equation 3 by 3: \(3 \times (80u + 110d) = 3 \times 13 \implies 240u + 330d = 39\) (Equation 5)
• Multiply Equation 4 by 4: \(4 \times (60u + 88d) = 4 \times 10 \implies 240u + 352d = 40\) (Equation 6)

Now, subtract Equation 5 from Equation 6:

\((240u + 352d) - (240u + 330d) = 40 - 39\)

\(240u + 352d - 240u - 330d = 1\)

\(22d = 1\)

\(d = \frac{1}{22}\)

Now substitute the value of \(d\) back into Equation 3 (or Equation 4) to find \(u\). Using Equation 3:

\(80u + 110 \left(\frac{1}{22}\right) = 13\)

\(80u + 5 = 13\)

\(80u = 13 - 5\)

\(80u = 8\)

\(u = \frac{8}{80} = \frac{1}{10}\)

Finding Boat Speed and Current Speed

We found \(u = \frac{1}{10}\) and \(d = \frac{1}{22}\). Let's substitute these back into our original definitions of \(u\) and \(d\):

• \(u = \frac{1}{b - c} = \frac{1}{10} \implies b - c = 10\) (Equation 7 - Upstream Speed)
• \(d = \frac{1}{b + c} = \frac{1}{22} \implies b + c = 22\) (Equation 8 - Downstream Speed)

We now have a simpler system of two linear equations with two variables \(b\) and \(c\).

To find \(c\) (speed of the current), subtract Equation 7 from Equation 8:

\((b + c) - (b - c) = 22 - 10\)

\(b + c - b + c = 12\)

\(2c = 12\)

\(c = \frac{12}{2}\)

\(c = 6\)

To find \(b\) (speed of the boat), substitute the value of \(c\) into Equation 7:

\(b - 6 = 10\)

\(b = 10 + 6\)

\(b = 16\)

Verification

Boat speed \(b = 16\) km/h, Current speed \(c = 6\) km/h.

• Upstream speed = \(b - c = 16 - 6 = 10\) km/h
• Downstream speed = \(b + c = 16 + 6 = 22\) km/h

Check Scenario 1: Time = \(\frac{80 \text{ km}}{10 \text{ km/h}} + \frac{110 \text{ km}}{22 \text{ km/h}} = 8 \text{ hours} + 5 \text{ hours} = 13\) hours. (Correct)

Check Scenario 2: Time = \(\frac{60 \text{ km}}{10 \text{ km/h}} + \frac{88 \text{ km}}{22 \text{ km/h}} = 6 \text{ hours} + 4 \text{ hours} = 10\) hours. (Correct)

The calculated speeds satisfy both conditions. The speed of the current is 6 km/h.

Revision Table: Rowing Speed Concepts

Additional Information: Solving Word Problems

Solving word problems like this often involves translating the problem into mathematical equations. Here are some tips:

• Read the problem carefully to understand all the given information and what is being asked.
• Define variables for the unknown quantities (like boat speed and current speed).
• Identify the relationships between the variables based on the problem's context (e.g., upstream speed, downstream speed).
• Use relevant formulas (like Time = Distance/Speed) to set up equations.
• Solve the system of equations. Substitution or elimination methods are commonly used.
• Check your answer by plugging the values back into the original conditions of the problem.

Problems involving upstream and downstream speeds are a classic application of setting up and solving systems of linear equations. Mastering this helps in various quantitative aptitude tests.