The ratio of time taken by a boat to cover a distance in downstream to that of in upstream is 3 : 5. If the boat covers 90 km in downstream in t hours and 72 km in upstream in (t+1) hours, then find downstream speed (in km/hr) of the boat.

The correct answer is 30 (or Option 4, 30 km/hr).

Let's break down the problem step-by-step to find the downstream speed of the boat.

Step 1: Understand the ratio of time taken for equal distance
Let $v_d$ be the downstream speed of the boat and $v_u$ be the upstream speed of the boat.
For a given constant distance, the time taken is inversely proportional to speed.
The problem states that the ratio of time taken in downstream to upstream for a certain distance is $3:5$.

Therefore, the ratio of downstream speed to upstream speed is the inverse of the time ratio:

$\frac{v_d}{v_u}=\frac{5}{3}$

This implies:

$v_u=\frac{3}{5}{v}_d$

Step 2: Express time taken in terms of downstream speed
The boat covers 90 km in downstream in $t$ hours.

$t=\frac{90}{v_d}$

The boat covers 72 km in upstream in $(t+1)$ hours.

$t+1=\frac{72}{v_u}$

Step 3: Substitute upstream speed in terms of downstream speed
Substitute $v_u=\frac{3}{5}{v}_d$ into the upstream time equation:

$t+1=\frac{72}{\frac{3}{5}{v}_d}=\frac{72\times 5}{3v_d}=\frac{24\times 5}{v_d}=\frac{120}{v_d}$

Step 4: Solve for downstream speed ($v_d$)
Now, subtract the expression for $t$ from the expression for $t+1$:

$(t+1)-t=\frac{120}{v_d}-\frac{90}{v_d}$

$1=\frac{120-90}{v_d}$

$1=\frac{30}{v_d}$

$v_d=30\text{km/hr}$

Thus, the downstream speed of the boat is 30 km/hr.