The respective ratio between speed of the boat upstream and speed of the boat downstream is 7 : 9. What is the speed of the boat in still water if it covers 48 km downstream in 2 h... The respective ratio between speed of the boat upstream and speed of the boat downstream is 7 : 9. What is the speed of the boat in still water if it covers 48 km downstream in 2 hours 40 minutes? (in km/h) Read more

Steps to Solve

1. Convert time to hours To find the speed, we first convert the time taken to travel downstream from hours and minutes to just hours.

• 2 hours 40 minutes can be converted as follows: $$ \text{Total hours} = 2 + \frac{40}{60} = 2 + \frac{2}{3} = \frac{8}{3} \text{ hours} $$

2. Calculate downstream speed To find the downstream speed, we use the formula: $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$ The distance covered downstream is 48 km, and the time is $\frac{8}{3}$ hours. $$ \text{Downstream speed} = \frac{48 \text{ km}}{\frac{8}{3} \text{ hours}} = 48 \times \frac{3}{8} = 18 \text{ km/h} $$

3. Determine the ratio of speeds From the problem, the ratio of speeds upstream to downstream is 7:9. Let the speed of the boat in still water be $x$. Therefore, we can express the upstream and downstream speeds as:

• Upstream speed = $x - c$ (where $c$ is the speed of the current)
• Downstream speed = $x + c$

Using the ratio: $$ \frac{x - c}{x + c} = \frac{7}{9} $$

4. Set up equations based on the ratios Cross-multiplying gives us: $$ 9(x - c) = 7(x + c) $$

Expanding and rearranging: $$ 9x - 9c = 7x + 7c $$

5. Solve for $c$ To isolate $c$, we rearrange the equation: $$ 9x - 7x = 9c + 7c $$ $$ 2x = 16c $$ $$ c = \frac{x}{8} $$

6. Substituting $c$ back to find $x$ Now substitute $c$ back into the downstream speed: $$ x + c = 18 \text{ km/h} $$ $$ x + \frac{x}{8} = 18 $$ Multiplying through by 8 to eliminate the fraction: $$ 8x + x = 144 $$ $$ 9x = 144 $$

7. Calculate the speed of the boat in still water Now, solve for $x$: $$ x = \frac{144}{9} = 16 \text{ km/h} $$