Rahim sets out to cross a forest. On the first day, he completes 1/10th of the journey. On the second day, he covers 2/3rd of the distance travelled the first day. He continues in this manner, alternating the days in which he travels 1/10th of the distance still to be covered, with days on which he travels 2/3 of the total distance already covered. At the end of seventh day, he finds that 22½ km more will see the end of his journey. How wide is the forest?

Options

A. `66" "2//3 km`
B. `100km`
C. `120km`
D. `150km`

Answer

To solve the problem step by step, we will analyze Rahim's journey across the forest based on the information provided.

### Step 1: Define the total distance of the forest
Let's assume the total distance of the forest is \( D \). To simplify calculations, we can choose \( D \) as a multiple of both 10 and 3. A suitable choice is \( D = 3000 \) km.

### Step 2: Calculate the distance covered on the first day
On the first day, Rahim covers \( \frac{1}{10} \) of the total distance:
\[ \text{Distance covered on Day 1} = \frac{1}{10} \times 3000 = 300 \text{ km} \]

### Step 3: Calculate the distance covered on the second day
On the second day, he covers \( \frac{2}{3} \) of the distance covered on the first day:
\[ \text{Distance covered on Day 2} = \frac{2}{3} \times 300 = 200 \text{ km} \]

### Step 4: Calculate the remaining distance after Day 2
After Day 2, the total distance covered is:
\[ \text{Total covered after Day 2} = 300 + 200 = 500 \text{ km} \]
The remaining distance is:
\[ \text{Remaining distance} = 3000 - 500 = 2500 \text{ km} \]

### Step 5: Calculate the distance covered on the third day
On the third day, he covers \( \frac{1}{10} \) of the remaining distance:
\[ \text{Distance covered on Day 3} = \frac{1}{10} \times 2500 = 250 \text{ km} \]

### Step 6: Calculate the total distance covered after Day 3
The total distance covered after Day 3 is:
\[ \text{Total covered after Day 3} = 500 + 250 = 750 \text{ km} \]

### Step 7: Calculate the remaining distance after Day 3
The remaining distance is:
\[ \text{Remaining distance} = 3000 - 750 = 2250 \text{ km} \]

### Step 8: Calculate the distance covered on the fourth day
On the fourth day, he covers \( \frac{2}{3} \) of the total distance covered so far:
\[ \text{Distance covered on Day 4} = \frac{2}{3} \times 750 = 500 \text{ km} \]

### Step 9: Calculate the total distance covered after Day 4
The total distance covered after Day 4 is:
\[ \text{Total covered after Day 4} = 750 + 500 = 1250 \text{ km} \]

### Step 10: Calculate the remaining distance after Day 4
The remaining distance is:
\[ \text{Remaining distance} = 3000 - 1250 = 1750 \text{ km} \]

### Step 11: Calculate the distance covered on the fifth day
On the fifth day, he covers \( \frac{1}{10} \) of the remaining distance:
\[ \text{Distance covered on Day 5} = \frac{1}{10} \times 1750 = 175 \text{ km} \]

### Step 12: Calculate the total distance covered after Day 5
The total distance covered after Day 5 is:
\[ \text{Total covered after Day 5} = 1250 + 175 = 1425 \text{ km} \]

### Step 13: Calculate the remaining distance after Day 5
The remaining distance is:
\[ \text{Remaining distance} = 3000 - 1425 = 1575 \text{ km} \]

### Step 14: Calculate the distance covered on the sixth day
On the sixth day, he covers \( \frac{2}{3} \) of the total distance covered so far:
\[ \text{Distance covered on Day 6} = \frac{2}{3} \times 1425 = 950 \text{ km} \]

### Step 15: Calculate the total distance covered after Day 6
The total distance covered after Day 6 is:
\[ \text{Total covered after Day 6} = 1425 + 950 = 2375 \text{ km} \]

### Step 16: Calculate the remaining distance after Day 6
The remaining distance is:
\[ \text{Remaining distance} = 3000 - 2375 = 625 \text{ km} \]

### Step 17: Calculate the distance covered on the seventh day
On the seventh day, he covers \( \frac{1}{10} \) of the remaining distance:
\[ \text{Distance covered on Day 7} = \frac{1}{10} \times 625 = 62.5 \text{ km} \]

### Step 18: Calculate the total distance covered after Day 7
The total distance covered after Day 7 is:
\[ \text{Total covered after Day 7} = 2375 + 62.5 = 2437.5 \text{ km} \]

### Step 19: Calculate the remaining distance after Day 7
The remaining distance is:
\[ \text{Remaining distance} = 3000 - 2437.5 = 562.5 \text{ km} \]

### Step 20: Relate the remaining distance to the given information
According to the problem, at the end of the seventh day, Rahim finds that 22.5 km more will see the end of his journey. Therefore:
\[ 562.5 \text{ km} = 22.5 \text{ km} \]

### Step 21: Solve for the total distance of the forest
From the above, we can set up the equation:
\[ \frac{9}{10} \times 625 = 22.5 \]
This implies:
\[ 625 = 22.5 \times \frac{10}{9} = 25 \]
Thus, the total distance of the forest \( D \) is:
\[ D = 3000 \text{ km} \Rightarrow \text{Total distance of the forest is } 120 \text{ km.} \]

### Final Answer
The width of the forest is **120 km**.

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