Example  4
 

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Jud takes the same amount of time to kayak \(12\) km downstream as he does to kayak \(6\) km upstream. The stream has a current of \(4\) km/h. If he maintains the same paddling speed in both directions, determine his still water paddling speed.



First, note that speed is equal to distance divided by time, or \(s = \frac{d}{t}\).

This formula can be manipulated to solve for time to give \(t = \frac{d}{s}\).

Step 1: Organize the information given.

Let \(x\) represent Jud’s speed in still water.

Complete a table to help organize the information:

Direction Distance (km) Speed (km/h) Time heart
Downstream \(12\) \(x + 4\)
\(\frac{12}{x + 4}\)
Upstream \(6\) \(x - 4\)
\(\frac{6}{x - 4}\)

Step 2: Write the equation.

The problem states that it takes the same amount of time for Jud to kayak the two distances. Therefore,

\[\frac{{12}}{{x + 4}} = \frac{6}{{x - 4}}\]

NPVs: \(x \ne\pm 4\)

\[\begin{align}
 \frac{{12}}{{x + 4}} &= \frac{6}{{x - 4}} \\
 12\left( {x - 4} \right) &= 6\left( {x + 4} \right) \\
 12x - 48 &= 6x + 24 \\
 6x &= 72 \\
 x &= 12 \\
 \end{align}\]

Verify for \(x = 12\).


Left Side Right Side
\[\begin{array}{r}
 \frac{{12}}{{x + 4}} \\
 \frac{{12}}{{\left( {12} \right) + 4}} \\
 \frac{{12}}{{16}} \\
 \frac{3}{4} \\
 \end{array}\]
 \[\begin{array}{l}
 \frac{6}{{x - 4}} \\
 \frac{6}{{\left( {12} \right) - 4}} \\
 \frac{6}{8} \\
 \frac{3}{4} \\
 \end{array}\]
LS = RS

Jud's still water paddling speed is \(12\) km/h.