Sure! Let's go step-by-step to find the average profit per shop from the given data.
### Step 1: Understand the Data
We are given the cumulative number of shops for different profit intervals. The intervals and cumulative counts are summarized as follows:
| Profit per Shop (Rs.) | No. of Shops |
|-----------------------|--------------|
| [tex]\(0-100\)[/tex] | 12 |
| [tex]\(0-200\)[/tex] | 30 |
| [tex]\(0-300\)[/tex] | 57 |
| [tex]\(0-400\)[/tex] | 77 |
| [tex]\(0-500\)[/tex] | 94 |
| [tex]\(0-600\)[/tex] | 100 |
### Step 2: Calculate the Number of Shops in Each Interval
Cumulative numbers tell us how many shops fall within or below a certain interval. To find the number of shops in each specific interval, we'll subtract the cumulative count at the lower end of the interval from the count at the higher end.
- Shops with profit [tex]\(0-100\)[/tex]: [tex]\(12\)[/tex] shops
- Shops with profit [tex]\(100-200\)[/tex]: [tex]\(30 - 12 = 18\)[/tex] shops
- Shops with profit [tex]\(200-300\)[/tex]: [tex]\(57 - 30 = 27\)[/tex] shops
- Shops with profit [tex]\(300-400\)[/tex]: [tex]\(77 - 57 = 20\)[/tex] shops
- Shops with profit [tex]\(400-500\)[/tex]: [tex]\(94 - 77 = 17\)[/tex] shops
- Shops with profit [tex]\(500-600\)[/tex]: [tex]\(100 - 94 = 6\)[/tex] shops
So, we have the following frequency distribution:
| Profit Interval (Rs.) | Midpoint (Rs.) | Frequency (Number of Shops) |
|------------------------|-----------------|-----------------------------|
| [tex]\(0-100\)[/tex] | 50 | 12 |
| [tex]\(100-200\)[/tex] | 150 | 18 |
| [tex]\(200-300\)[/tex] | 250 | 27 |
| [tex]\(300-400\)[/tex] | 350 | 20 |
| [tex]\(400-500\)[/tex] | 450 | 17 |
| [tex]\(500-600\)[/tex] | 550 | 6 |
### Step 3: Calculate the Total Profit
We use the midpoint of each interval as the representative value for the profits of all shops within that interval. The total profit is the sum of the products of the midpoint and corresponding frequency for each interval.
[tex]\[ \text{Total Profit} = (50 \times 12) + (150 \times 18) + (250 \times 27) + (350 \times 20) + (450 \times 17) + (550 \times 6) \][/tex]
[tex]\[
\text{Total Profit} = 600 + 2700 + 6750 + 7000 + 7650 + 3300 = 28000 \text{ Rs.}
\][/tex]
### Step 4: Calculate the Average Profit
The average profit per shop is obtained by dividing the total profit by the total number of shops.
[tex]\[
\text{Total Shops} = 100
\][/tex]
[tex]\[
\text{Average Profit per Shop} = \frac{\text{Total Profit}}{\text{Total Shops}} = \frac{28000}{100} = 280 \text{ Rs.}
\][/tex]
### Conclusion
Thus, the average profit per shop is [tex]\( \boxed{280 \text{ Rs.}} \)[/tex].
