Certain number of people work in retail, online and door to door stores. There are only three type stores and each people works in one or more store. 72% of people were in retail store and people working in only door to door store was 1 36 th of people working in retail store. Number of people working in both door to door store and online store but not in retail store are 55. People working in only online store are 65 more than the people working in only door to door store. Number of people working in only retail store is 160 3 % more than number of people working in online store.
What is the total number of people working in all the stores?

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Accepted Answer

**Solution:**

1. Let the total number of people be $N$.

2. Given, 72\% of people work in retail store, so:
   $$ \text{Retail} = 0.72N $$

3. Number of people working only in door-to-door store is $\frac{1}{36}$ of retail store workers:
   $$ \text{Only Door-to-Door} = \frac{1}{36} \times 0.72N = 0.02N $$

4. Number of people working in both door-to-door and online stores but not in retail store is 55.

5. Number of people working only in online store is 65 more than those working only in door-to-door store:
   $$ \text{Only Online} = 0.02N + 65 $$

6. Number of people working only in retail store is 160\% more than those working only in online store. 160\% more means $1 + \frac{160}{100} = 2.6$ times:
   $$ \text{Only Retail} = 2.6 \times \text{Only Online} = 2.6 (0.02N + 65) $$

7. Total people working in retail store is 72\% of $N$, which includes people working only in retail, people working in retail and online, people working in retail and door-to-door, and people working in all three stores.

Let the number of people working in both retail and online but not door-to-door be $x$, and the number working in both retail and door-to-door but not online be $y$, and those working in all three stores be $z$.

Then:
   $$ \text{Retail} = \text{Only Retail} + x + y + z = 0.72N $$

8. Similarly, the number of people working in door-to-door store is:
   $$ \text{Only Door-to-Door} + y + 55 + z $$

Here, 55 is the number working in door-to-door and online but not retail.

9. The number of people working in online store is:
   $$ \text{Only Online} + x + 55 + z $$

10. The total number of people $N$ is the sum of all disjoint groups:

$$ N = \text{Only Retail} + \text{Only Online} + \text{Only Door-to-Door} + x + y + 55 + z $$

11. From step 7:

$$ x + y + z = 0.72N - \text{Only Retail} $$

12. Substitute $ \text{Only Retail} = 2.6 (0.02N + 65) = 2.6 (0.02N) + 2.6 \times 65 = 0.052N + 169 $

13. So,

$$ x + y + z = 0.72N - (0.052N + 169) = 0.668N - 169 $$

14. Total people $N$ is:

$$ N = (0.052N + 169) + (0.02N + 65) + 0.02N + x + y + 55 + z $$

Group terms:

$$ N = 0.052N + 169 + 0.02N + 65 + 0.02N + (x + y + z) + 55 $$

$$ N = (0.052N + 0.02N + 0.02N) + (169 + 65 + 55) + (x + y + z) $$

$$ N = 0.092N + 289 + (x + y + z) $$

15. Substitute $ x + y + z = 0.668N - 169 $ from step 13:

$$ N = 0.092N + 289 + 0.668N - 169 $$

$$ N = (0.092N + 0.668N) + (289 - 169) $$

$$ N = 0.76N + 120 $$

16. Rearranging:

$$ N - 0.76N = 120 $$

$$ 0.24N = 120 $$

$$ N = \frac{120}{0.24} = 500 $$

17. Check for consistency:

- Only door-to-door: $0.02 \times 500 = 10$
    - Only online: $10 + 65 = 75$
    - Only retail: $2.6 \times 75 = 195$
    - Retail total: $0.72 \times 500 = 360$
    - Sum of $x + y + z = 360 - 195 = 165$

Total sum:

$$ 195 + 75 + 10 + 165 + 55 = 500 $$

(Note: 55 is given for door-to-door and online only)

The numbers add up correctly.

**Answer:** 500

Since 500 is not among the options, re-examine the calculation carefully.

**Re-examining step 16:**

We had:

$$ N = 0.76N + 120 \implies N - 0.76N = 120 \implies 0.24N = 120 \implies N = 500 $$

But the options include 360, 300, 250, 400, 500.

Let's check if the problem states the number of people working only in retail store is 160\% more than the number working in online store or 160\% more than the number working only in online store.

The problem states: "Number of people working in only retail store is 160\% more than number of people working in online store."

If it means only online store, then calculation is correct.

Therefore, the total number of people is $500$.

**Answer:** 500

**Final Note:** Since 500 is an option, the correct answer is 500.

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