To solve for the speed of the current, let's go through the steps systematically.
1. Define the Variables:
- Speed of the boat in still water: [tex]\( 10 \)[/tex] miles per hour
- Distance traveled downstream: [tex]\( 6 \)[/tex] miles
- Distance traveled upstream: [tex]\( 4 \)[/tex] miles
- Speed of the current: [tex]\( x \)[/tex] miles per hour
2. Formulate the Time Equations:
- When traveling downstream, the effective speed of the boat is the speed of the boat in still water plus the speed of the current: [tex]\( 10 + x \)[/tex].
- Time taken to travel downstream [tex]\( t_1 = \frac{6}{10 + x} \)[/tex]
- When traveling upstream, the effective speed of the boat is the speed of the boat in still water minus the speed of the current: [tex]\( 10 - x \)[/tex].
- Time taken to travel upstream [tex]\( t_2 = \frac{4}{10 - x} \)[/tex]
3. Set Up the Rational Equation:
Since the times taken to travel the given distances downstream and upstream are the same, we equate the two times:
[tex]\[
\frac{6}{10 + x} = \frac{4}{10 - x}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Cross-multiplying the equation gives:
[tex]\[
6 \cdot (10 - x) = 4 \cdot (10 + x)
\][/tex]
- Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[
60 - 6x = 40 + 4x
\][/tex]
[tex]\[
60 - 40 = 4x + 6x
\][/tex]
[tex]\[
20 = 10x
\][/tex]
[tex]\[
x = 2
\][/tex]
Therefore, the speed of the current is [tex]\( \boxed{2} \)[/tex] miles per hour.
