A boat covers 24 km upstream and 36 km downstream in 10 hours, and 36 km upstream and 24 km downstream in 12 hours. The speed of the current is:

The correct option is 25/8 km/h.

Step-by-step explanation:

Let the upstream speed of the boat be $u$ km/h and the downstream speed of the boat be $v$ km/h.

We know that:

$\text{Time}=\frac{\text{Distance}}{\text{Speed}}$

Step 1: Set up equations based on the given information

First scenario: The boat covers 24 km upstream and 36 km downstream in 10 hours:

$\frac{24}{u}+\frac{36}{v}=10$

Second scenario: The boat covers 36 km upstream and 24 km downstream in 12 hours:

$\frac{36}{u}+\frac{24}{v}=12$

Step 2: Solve for upstream speed ($u$) and downstream speed ($v$)

Let $a=\frac{1}{u}$ and $b=\frac{1}{v}$. Substituting these into the equations gives:

$24a+36b=10$   (Equation 1)

$36a+24b=12$   (Equation 2)

Multiply Equation 1 by 3:

$72a+108b=30$

Multiply Equation 2 by 2:

$72a+48b=24$

Subtracting the second equation from the first equation:

$60b=6$

$b=\frac{6}{60}=\frac{1}{10}$

Therefore, the downstream speed is:

$v=\frac{1}{b}=10$ km/h

Now substitute $b=\frac{1}{10}$ into Equation 1:

$24a+36\left(\frac{1}{10}\right)=10$

$24a+3.6=10$

$24a=6.4$

$a=\frac{6.4}{24}=\frac{64}{240}=\frac{4}{15}$

Therefore, the upstream speed is:

$u=\frac{1}{a}=\frac{15}{4}$ km/h

Step 3: Calculate the speed of the current

Let $x$ be the speed of the boat in still water and $y$ be the speed of the current.

We know:

Downstream speed: $v=x+y=10$

Upstream speed: $u=x-y=\frac{15}{4}$

Subtracting the upstream equation from the downstream equation gives:

$(x+y)-(x-y)=10-\frac{15}{4}$

$2y=\frac{40}{4}-\frac{15}{4}$

$2y=\frac{25}{4}$

$y=\frac{25}{8}$

Hence, the speed of the current is 25/8 km/h.