How to Solve a Distance, Rate & Time Problem Using a Rational ... - Free Printable

Problem Statement:

Dimitri can row his boat downstream 15 miles in the same amount of time as it takes him to row upstream 8 miles. The current in the river is 2 mph. How fast can Dimitri row in still water?

Solution:

#### Step 1: Define Variables
Let:
- \( x \): Speed of Dimitri in still water (in mph).
- Current speed = 2 mph.

When rowing downstream, the effective speed is the sum of Dimitri's speed and the current speed:
\[ \text{Downstream speed} = x + 2 \]

When rowing upstream, the effective speed is the difference between Dimitri's speed and the current speed:
\[ \text{Upstream speed} = x - 2 \]

#### Step 2: Use the Relationship Between Distance, Rate, and Time
The formula for time is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Rate}} \]

- Time to row downstream 15 miles:
\[ t_{\text{down}} = \frac{15}{x + 2} \]

- Time to row upstream 8 miles:
\[ t_{\text{up}} = \frac{8}{x - 2} \]

According to the problem, the time taken to row downstream 15 miles is the same as the time taken to row upstream 8 miles:
\[ t_{\text{down}} = t_{\text{up}} \]

Thus:
\[ \frac{15}{x + 2} = \frac{8}{x - 2} \]

#### Step 3: Solve the Equation
Cross-multiply to eliminate the fractions:
\[ 15(x - 2) = 8(x + 2) \]

Expand both sides:
\[ 15x - 30 = 8x + 16 \]

Rearrange the equation to isolate \( x \):
\[ 15x - 8x = 16 + 30 \]
\[ 7x = 46 \]

Solve for \( x \):
\[ x = \frac{46}{7} \]

#### Step 4: Simplify the Result
\[ x \approx 6.57 \text{ mph} \]

#### Final Answer:
\[ \boxed{6.6} \]

Explanation:

- The key to solving this problem is understanding how the current affects Dimitri's speed when rowing downstream and upstream.
- By setting the times equal and solving the resulting equation, we find Dimitri's speed in still water.
- The final answer, \( x = \frac{46}{7} \), is approximately 6.57 mph, which rounds to 6.6 mph.

Thus, Dimitri can row at approximately 6.6 mph in still water.