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

View step-by-step help

Home

Math Problems

Algebra 1

Write exponential functions: word problems

Full solution

Q. Naira paddled

4.5 \mathrm{~km}

with the current in the same amount of time as it took her to paddle

3 \mathrm{~km}

against the current. Naira paddled at an average rate of

5 \frac{\mathrm{km}}{\mathrm{h}}

relative to the water each way. Assume the speed of the current was constant.

\newline

What was the speed of the current?

\newline

\square \frac{\mathrm{km}}{\mathrm{h}}

Denote current speed: Let's denote the speed of the current as $c$ (km/h). Naira's speed relative to the water is $5$ km/h. When paddling with the current, her effective speed is $(5 + c)$ km/h, and when paddling against the current, her effective speed is $(5 - c)$ km/h. We know that the time taken to paddle in both directions is the same. We can use the formula $\text{time} = \frac{\text{distance}}{\text{speed}}$ to set up our equation.
Calculate time with current: First, let's write down the time it takes Naira to paddle with the current: $\text{time\_with\_current} = \frac{\text{distance\_with\_current}}{\text{speed\_with\_current}}$ . This gives us $\text{time\_with\_current} = \frac{4.5 \, \text{km}}{(5 \, \text{km/h} + c)}$ .
Calculate time against current: Now, let's write down the time it takes Naira to paddle against the current: $\text{time\_against\_current} = \frac{\text{distance\_against\_current}}{\text{speed\_against\_current}}$ . This gives us $\text{time\_against\_current} = \frac{3 \, \text{km}}{(5 \, \text{km/h} - c)}$ .
Set up equation: Since the times are equal, we can set the two expressions equal to each other: $\frac{4.5 \, \text{km}}{(5 \, \text{km/h} + c)} = \frac{3 \, \text{km}}{(5 \, \text{km/h} - c)}$ .
Cross-multiply to solve: To solve for $c$ , we cross-multiply: $(4.5 \, \text{km}) \times (5 \, \text{km/h} - c) = (3 \, \text{km}) \times (5 \, \text{km/h} + c)$ .
Expand and simplify equation: Expanding both sides of the equation gives us: $4.5 \times 5 - 4.5c = 3 \times 5 + 3c$ .
Combine like terms: Simplify the equation: $22.5 - 4.5c = 15 + 3c$ .
Divide to solve for c: Now, let's combine like terms by adding $4.5c$ to both sides and subtracting $15$ from both sides: $22.5 - 15 = 3c + 4.5c$ .
Final result: This simplifies to: $7.5 = 7.5c$ .
Final result: This simplifies to: $7.5 = 7.5c$ . Finally, divide both sides by $7.5$ to solve for $c$ : $c = \frac{7.5}{7.5}.$
Final result: This simplifies to: $7.5 = 7.5c$ . Finally, divide both sides by $7.5$ to solve for $c$ : $c = \frac{7.5}{7.5}$ . This gives us: $c = 1$ km/h.