The ratio of time taken by a boat to cover a certain distance downstream to the time taken upstream is $2 : 3$. The boat covers $108 ext{ km}$ downstream in $d$ hours and $120 ext{ km}$ upstream in $d + 4$ hours. What is the upstream speed of the boat in km/hr?

Question

The ratio of time taken by a boat to cover a certain distance downstream to the time taken upstream is $2 : 3$. The boat covers $108 	ext{ km}$ downstream in $d$ hours and $120 	ext{ km}$ upstream in $d + 4$ hours. What is the upstream speed of the boat in km/hr?

Accepted Answer

**Solution:**

1. Let the time taken downstream be $2x$ hours and upstream be $3x$ hours, since the ratio of time downstream to upstream is $2 : 3$.

2. Given that downstream distance is $108 	ext{ km}$ covered in $d$ hours, and upstream distance is $120 	ext{ km}$ covered in $d + 4$ hours.

3. From the ratio, $d = 2x$ and $d + 4 = 3x$.

4. Substitute $d = 2x$ into $d + 4 = 3x$:
   $$2x + 4 = 3x \ 4 = 3x - 2x = x$$

5. So, $x = 4$.

6. Therefore, downstream time $d = 2x = 8$ hours, and upstream time $d + 4 = 12$ hours.

7. Calculate downstream speed:
   $$	ext{Downstream speed} = \frac{108}{8} = 13.5 	ext{ km/hr}$$

8. Calculate upstream speed:
   $$	ext{Upstream speed} = \frac{120}{12} = 10 	ext{ km/hr}$$

9. However, the question asks for upstream speed, but the options do not include 10 km/hr. Let's re-check the interpretation.

**Re-examining the problem:**

- The ratio of times downstream to upstream is $2 : 3$.
- Given downstream distance = 108 km in $d$ hours.
- Given upstream distance = 120 km in $d + 4$ hours.

From the ratio:
$$\frac{d}{d + 4} = \frac{2}{3}$$

Cross-multiplied:
$$3d = 2(d + 4) \ 3d = 2d + 8 \ d = 8$$

So,
- Downstream time $d = 8$ hours
- Upstream time $d + 4 = 12$ hours

Calculate speeds:
- Downstream speed = $\frac{108}{8} = 13.5$ km/hr
- Upstream speed = $\frac{120}{12} = 10$ km/hr

Since 10 km/hr is not in options, let's check if the question expects upstream speed as per ratio or if the ratio is time downstream to upstream or vice versa.

The problem states: "The ratio of time taken by a boat to cover a distance in downstream to that of in upstream is $2 : 3$".

So time downstream : time upstream = 2 : 3

Given distances and times:
- Downstream distance = 108 km in $d$ hours
- Upstream distance = 120 km in $d + 4$ hours

From ratio:
$$\frac{d}{d + 4} = \frac{2}{3}$$

Solving gives $d = 8$ hours.

Speeds:
- Downstream speed = $108 / 8 = 13.5$ km/hr
- Upstream speed = $120 / 12 = 10$ km/hr

Since 10 km/hr is not an option, let's consider if the ratio is reversed or if the question expects the speed ratio.

Alternatively, consider the ratio of speeds:

Speed downstream = $\frac{108}{d}$
Speed upstream = $\frac{120}{d + 4}$

Time ratio is $2 : 3$ so speed ratio is inverse, i.e., $3 : 2$.

Check if:
$$\frac{108}{d} : \frac{120}{d + 4} = 3 : 2$$

Cross multiply:
$$\frac{108}{d} 	imes 2 = \frac{120}{d + 4} 	imes 3$$

$$\frac{216}{d} = \frac{360}{d + 4}$$

Cross multiply:
$$216(d + 4) = 360 d$$

$$216 d + 864 = 360 d$$

$$864 = 360 d - 216 d = 144 d$$

$$d = \frac{864}{144} = 6$$

So,
- Downstream time $d = 6$ hours
- Upstream time $d + 4 = 10$ hours

Speeds:
- Downstream speed = $\frac{108}{6} = 18$ km/hr
- Upstream speed = $\frac{120}{10} = 12$ km/hr

Upstream speed is $12$ km/hr, which matches option (d).

**Answer:** $12$ km/hr

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