Ram went on a 10 mile drive. He started with a certain speed and after covering each mile, his speed is decreased by 20% for the next mile. If he took 5 minutes to cover the first 5 mile of the drive, what is the approximate time taken by him to cover the next 5 miles?

Answer

To solve the problem step by step, we will calculate the time taken by Ram to cover the next 5 miles after the first 5 miles, where his speed decreases by 20% after each mile.

### Step 1: Determine the initial speed
Let Ram's initial speed be \( S \) miles per minute. Since he took 5 minutes to cover the first 5 miles, we can find his speed using the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{5 \text{ miles}}{5 \text{ minutes}} = 1 \text{ mile per minute} \]
Thus, \( S = 1 \) mile per minute.

### Step 2: Calculate the speed for each mile
After each mile, his speed decreases by 20%. Therefore, the speed for each mile can be calculated as follows:
- Speed for the 1st mile: \( S = 1 \) mile/min
- Speed for the 2nd mile: \( S \times 0.8 = 1 \times 0.8 = 0.8 \) miles/min
- Speed for the 3rd mile: \( S \times 0.8^2 = 1 \times 0.64 = 0.64 \) miles/min
- Speed for the 4th mile: \( S \times 0.8^3 = 1 \times 0.512 = 0.512 \) miles/min
- Speed for the 5th mile: \( S \times 0.8^4 = 1 \times 0.4096 = 0.4096 \) miles/min

### Step 3: Calculate the time taken for each mile
Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can find the time taken for each of the next 5 miles:
- Time for the 6th mile:
\[ t_6 = \frac{1}{0.32768} \text{ minutes} \quad (\text{Speed for 6th mile} = S \times 0.8^5 = 0.32768 \text{ miles/min}) \]
- Time for the 7th mile:
\[ t_7 = \frac{1}{0.262144} \text{ minutes} \quad (\text{Speed for 7th mile} = S \times 0.8^6 = 0.262144 \text{ miles/min}) \]
- Time for the 8th mile:
\[ t_8 = \frac{1}{0.2097152} \text{ minutes} \quad (\text{Speed for 8th mile} = S \times 0.8^7 = 0.2097152 \text{ miles/min}) \]
- Time for the 9th mile:
\[ t_9 = \frac{1}{0.16777216} \text{ minutes} \quad (\text{Speed for 9th mile} = S \times 0.8^8 = 0.16777216 \text{ miles/min}) \]
- Time for the 10th mile:
\[ t_{10} = \frac{1}{0.134217728} \text{ minutes} \quad (\text{Speed for 10th mile} = S \times 0.8^9 = 0.134217728 \text{ miles/min}) \]

### Step 4: Calculate the total time for the next 5 miles
The total time for the next 5 miles is:
\[ \text{Total time} = t_6 + t_7 + t_8 + t_9 + t_{10} \]
Calculating each term:
\[ t_6 \approx 3.05 \text{ min}, \quad t_7 \approx 3.81 \text{ min}, \quad t_8 \approx 4.77 \text{ min}, \quad t_9 \approx 5.96 \text{ min}, \quad t_{10} \approx 7.46 \text{ min} \]
Adding these gives:
\[ \text{Total time} \approx 3.05 + 3.81 + 4.77 + 5.96 + 7.46 \approx 25.05 \text{ minutes} \]

### Final Answer
The approximate time taken by Ram to cover the next 5 miles is **25 minutes**.

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