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Naira paddled 4.5km with the current in the same amount of time as it took her to paddle 3km against the current. Naira paddled at an average rate of 5(km)/(h) relative to the water each way. Assume the speed of the current was constant. What was the speed of the current? (km)/(h)

Naira paddled 4.5 km 4.5 \mathrm{~km} with the current in the same amount of time as it took her to paddle 3 km 3 \mathrm{~km} against the current. Naira paddled at an average rate of 5kmh 5 \frac{\mathrm{km}}{\mathrm{h}} relative to the water each way. Assume the speed of the current was constant.\newlineWhat was the speed of the current?\newline□kmh\square \frac{\mathrm{km}}{\mathrm{h}}

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Q. Naira paddled 4.5 km 4.5 \mathrm{~km} with the current in the same amount of time as it took her to paddle 3 km 3 \mathrm{~km} against the current. Naira paddled at an average rate of 5kmh 5 \frac{\mathrm{km}}{\mathrm{h}} relative to the water each way. Assume the speed of the current was constant.\newlineWhat was the speed of the current?\newline□kmh\square \frac{\mathrm{km}}{\mathrm{h}}
  1. Denote current speed: Let's denote the speed of the current as cc (km/h). Naira's speed relative to the water is 55 km/h. When paddling with the current, her effective speed is (5+c)(5 + c) km/h, and when paddling against the current, her effective speed is (5−c)(5 - c) km/h. We know that the time taken to paddle in both directions is the same. We can use the formula time=distancespeed\text{time} = \frac{\text{distance}}{\text{speed}} to set up our equation.
  2. Calculate time with current: First, let's write down the time it takes Naira to paddle with the current: time_with_current=distance_with_currentspeed_with_current\text{time\_with\_current} = \frac{\text{distance\_with\_current}}{\text{speed\_with\_current}}. This gives us time_with_current=4.5 km(5 km/h+c)\text{time\_with\_current} = \frac{4.5 \, \text{km}}{(5 \, \text{km/h} + c)}.
  3. Calculate time against current: Now, let's write down the time it takes Naira to paddle against the current: time_against_current=distance_against_currentspeed_against_current\text{time\_against\_current} = \frac{\text{distance\_against\_current}}{\text{speed\_against\_current}}. This gives us time_against_current=3 km(5 km/h−c)\text{time\_against\_current} = \frac{3 \, \text{km}}{(5 \, \text{km/h} - c)}.
  4. Set up equation: Since the times are equal, we can set the two expressions equal to each other: 4.5 km(5 km/h+c)=3 km(5 km/h−c)\frac{4.5 \, \text{km}}{(5 \, \text{km/h} + c)} = \frac{3 \, \text{km}}{(5 \, \text{km/h} - c)}.
  5. Cross-multiply to solve: To solve for cc, we cross-multiply: (4.5 km)×(5 km/h−c)=(3 km)×(5 km/h+c)(4.5 \, \text{km}) \times (5 \, \text{km/h} - c) = (3 \, \text{km}) \times (5 \, \text{km/h} + c).
  6. Expand and simplify equation: Expanding both sides of the equation gives us: 4.5×5−4.5c=3×5+3c4.5 \times 5 - 4.5c = 3 \times 5 + 3c.
  7. Combine like terms: Simplify the equation: 22.5−4.5c=15+3c22.5 - 4.5c = 15 + 3c.
  8. Divide to solve for c: Now, let's combine like terms by adding 4.5c4.5c to both sides and subtracting 1515 from both sides: 22.5−15=3c+4.5c22.5 - 15 = 3c + 4.5c.
  9. Final result: This simplifies to: 7.5=7.5c7.5 = 7.5c.
  10. Final result: This simplifies to: 7.5=7.5c7.5 = 7.5c. Finally, divide both sides by 7.57.5 to solve for cc: c=7.57.5.c = \frac{7.5}{7.5}.
  11. Final result: This simplifies to: 7.5=7.5c7.5 = 7.5c. Finally, divide both sides by 7.57.5 to solve for cc: c=7.57.5c = \frac{7.5}{7.5}. This gives us: c=1c = 1 km/h.

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