In 500 small scale units, the return on investment ranged from 0 to 30 per cent, no 
unit sustaining any loss. Five per cent of the units had returns ranging from zero per cent 
to 5 per cent, 15 per cent of the units earned returns between 5 per cent and 10 per cent. 
The median rate of return was 15 per cent and the upper quartile was 20 per cent. The 
uppermost layer of returns of 25—30 per cent was earned by 50 units. Put this information 
in the form of a frequency table and find the rate of return around which there is 
maximum concentration of units.

Let's break down the problem and organize the given data into a frequency table.

### Step 1: Analyze the Data
1. **Total Units**: 500 units.
2. **Return Range**: 0% to 30%.
3. **Frequency Distribution**:
   - 5% of units (0% to 5%): \( 5\% \times 500 = 25 \) units.
   - 15% of units (5% to 10%): \( 15\% \times 500 = 75 \) units.
   - The median rate of return is 15%.
   - The upper quartile is 20%.
   - 50 units earned returns between 25% and 30%.

### Step 2: Construct the Frequency Table

Given the above data, we can construct the frequency table as follows:

| Return Range (%) | Number of Units | Cumulative Frequency |
|------------------|-----------------|----------------------|
| 0 - 5            | 25              | 25                   |
| 5 - 10           | 75              | 100                  |
| 10 - 15          | \( x_1 \)       | \( 100 + x_1 \)      |
| 15 - 20          | \( x_2 \)       | \( 100 + x_1 + x_2 \)|
| 20 - 25          | \( x_3 \)       | \( 100 + x_1 + x_2 + x_3 \) |
| 25 - 30          | 50              | 500                  |

### Step 3: Solve for the Unknowns

We know that the median return is 15%, meaning that half of the units (250) earned less than or equal to 15%. Therefore, the cumulative frequency at the 15% mark should be 250 units. 

So, \( 100 + x_1 = 250 \) implies:
\[
x_1 = 250 - 100 = 150 \text{ units in the range } 10\%-15\%.
\]

The upper quartile is at 20%, meaning that 75% of the units (375 units) earned less than or equal to 20%.

So, \( 100 + 150 + x_2 = 375 \) implies:
\[
x_2 = 375 - 250 = 125 \text{ units in the range } 15\%-20\%.
\]

Finally, there are 50 units in the range 25%-30%, so:
\[
x_3 = 500 - (100 + 150 + 125 + 50) = 75 \text{ units in the range } 20\%-25\%.
\]

### Final Frequency Table

| Return Range (%) | Number of Units | Cumulative Frequency |
|------------------|-----------------|----------------------|
| 0 - 5            | 25              | 25                   |
| 5 - 10           | 75              | 100                  |
| 10 - 15          | 150             | 250                  |
| 15 - 20          | 125             | 375                  |
| 20 - 25          | 75              | 450                  |
| 25 - 30          | 50              | 500                  |

### Step 4: Identify Maximum Concentration

The range with the maximum number of units (maximum concentration) is **10% to 15%** with 150 units.

---

Do you want more details or have any questions?

Here are five related questions:
1. How would the frequency table change if 10% of units had returns from 0% to 5%?
2. What is the interquartile range for the return on investment?
3. How would you calculate the mode of this dataset?
4. What would be the impact if the number of units in the 25%-30% range doubled?
5. How does the cumulative frequency help in understanding the distribution?

**Tip:** When analyzing distributions, the cumulative frequency helps in understanding the spread and concentration of data, especially around key statistical points like the median and quartiles.

Explore a detailed analysis of return on investment with a frequency distribution. This problem involves analyzing returns ranging from 0% to 30%, with concentration points such as the median and quartiles. Suitable for undergraduate-level study.

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