Solving Boat and Stream Speed Problems

Boat and stream problems often involve setting up equations based on the time taken to cover distances upstream and downstream. The key is understanding how the speed of the stream affects the boat's speed in different directions.

Understanding Upstream and Downstream Motion

• Upstream: When the boat travels against the current, the speed of the stream reduces the boat's effective speed. The speed is the speed of the boat in still water minus the speed of the stream.
• Downstream: When the boat travels with the current, the speed of the stream increases the boat's effective speed. The speed is the speed of the boat in still water plus the speed of the stream.

Let's define the variables:

• Let $B$ be the speed of the boat in still water (in km/hr).
• Let $S$ be the speed of the stream (in km/hr).
• Upstream speed $= B - S$ km/hr.
• Downstream speed $= B + S$ km/hr.

The general formula connecting 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. A boat covers 20 km upstream and 60 km downstream in a total of 5 hours.
2. A boat covers 30 km upstream and 80 km downstream in a total of 7 hours.

Using the time formula, we can write the equations:

• For scenario 1: Time upstream + Time downstream = 5 hours
• $\frac{20}{B - S} + \frac{60}{B + S} = 5 \quad (\text{Equation 1})$

• For scenario 2: Time upstream + Time downstream = 7 hours
• $\frac{30}{B - S} + \frac{80}{B + S} = 7 \quad (\text{Equation 2})$

Solving the System of Equations for Boat Speed

These are a system of equations with reciprocals. To simplify, let's make substitutions:

• Let $u = \frac{1}{B - S}$ (which is $\frac{1}{\text{Upstream Speed}}$)
• Let $v = \frac{1}{B + S}$ (which is $\frac{1}{\text{Downstream Speed}}$)

Substituting $u$ and $v$ into our equations:

• Equation 1 becomes: $20u + 60v = 5$
• We can simplify this by dividing by 5: $4u + 12v = 1 \quad (\text{Equation 3})$

• Equation 2 becomes: $30u + 80v = 7 \quad (\text{Equation 4})$

Now we have a linear system in terms of $u$ and $v$. We can solve this using methods like elimination or substitution.

Let's use the elimination method. Multiply Equation 3 by a number that makes the coefficient of $u$ equal to the coefficient of $u$ in Equation 4. The ratio is $\frac{30}{4} = 7.5$.

• Multiply Equation 3 by 7.5: $7.5 \times (4u + 12v) = 7.5 \times 1$
• This gives: $30u + 90v = 7.5 \quad (\text{Equation 5})$

Now subtract Equation 4 from Equation 5:

$(30u + 90v) - (30u + 80v) = 7.5 - 7$

$30u + 90v - 30u - 80v = 0.5$

$10v = 0.5$

$v = \frac{0.5}{10} = 0.05$ or $v = \frac{1}{20}$

Now substitute the value of $v$ back into Equation 3 ($4u + 12v = 1$):

$4u + 12 \left(\frac{1}{20}\right) = 1$

$4u + \frac{12}{20} = 1$

$4u + \frac{3}{5} = 1$

$4u = 1 - \frac{3}{5}$

$4u = \frac{5 - 3}{5}$

$4u = \frac{2}{5}$

$u = \frac{2}{5 \times 4} = \frac{2}{20} = \frac{1}{10}$

So, we have $u = \frac{1}{10}$ and $v = \frac{1}{20}$.

Now, recall the substitutions we made:

• $u = \frac{1}{B - S} \implies \frac{1}{B - S} = \frac{1}{10} \implies B - S = 10 \quad (\text{Equation 6})$
• $v = \frac{1}{B + S} \implies \frac{1}{B + S} = \frac{1}{20} \implies B + S = 20 \quad (\text{Equation 7})$

We now have a simple system of two linear equations with two variables $B$ and $S$.

To find the speed of the boat in still water ($B$), we can add Equation 6 and Equation 7:

$(B - S) + (B + S) = 10 + 20$

$B - S + B + S = 30$

$2B = 30$

$B = \frac{30}{2}$

$B = 15$ km/hr

To find the speed of the stream ($S$), substitute $B=15$ into Equation 7:

$15 + S = 20$

$S = 20 - 15$

$S = 5$ km/hr

The speed of the boat in still water is 15 km/hr.

Verification

Let's check if these speeds satisfy the original conditions:

• Upstream speed = $B - S = 15 - 5 = 10$ km/hr
• Downstream speed = $B + S = 15 + 5 = 20$ km/hr

• Scenario 1: 20 km upstream and 60 km downstream
• Time = $\frac{20}{10} + \frac{60}{20} = 2 + 3 = 5$ hours. This matches the given time.

• Scenario 2: 30 km upstream and 80 km downstream
• Time = $\frac{30}{10} + \frac{80}{20} = 3 + 4 = 7$ hours. This also matches the given time.

The calculated speeds are correct.

Revision Table: Boat and Stream Concepts

Reviewing the core formulas for boat and stream problems:

Additional Information: Solving Linear Equations

The problem reduced to solving a system of two linear equations with two variables. Common methods include:

• Substitution: Solve one equation for one variable and substitute that expression into the other equation.
• Elimination: Multiply one or both equations by constants so that the coefficients of one variable are opposites, then add the equations to eliminate that variable.

In this problem, we used a combination of substitution (for $u$ and $v$) and elimination (for $u$ and $v$, and then for $S$). Understanding these methods is crucial for many quantitative problems.