Given the data of items sold by a store over five days, find the difference between the average number of items B sold on Tuesday and Thursday and the average number of items A sold on Thursday and Friday.

| Days | Items A | % of items B | % of items C |
|-----------|---------|--------------|--------------|
| Monday | 240 | 32\% | 20\% |
| Tuesday | 320 | 48\% | 12\% |
| Wednesday | 420 | 45\% | 20\% |
| Thursday | 360 | 56\% | 20\% |
| Friday | 340 | 22\% | 10\% |

Question

Given the data of items sold by a store over five days, find the difference between the average number of items B sold on Tuesday and Thursday and the average number of items A sold on Thursday and Friday.

| Days      | Items A | % of items B | % of items C |
|-----------|---------|--------------|--------------|
| Monday    | 240     | 32\%         | 20\%         |
| Tuesday   | 320     | 48\%         | 12\%         |
| Wednesday | 420     | 45\%         | 20\%         |
| Thursday  | 360     | 56\%         | 20\%         |
| Friday    | 340     | 22\%         | 10\%         |

Accepted Answer

**Solution:**

1. Calculate the number of items B sold on Tuesday and Thursday:
   - On Tuesday, total items A sold = 320, percentage of items B = 48\%
     $$	ext{Items B on Tuesday} = \frac{48}{100} 	imes 320 = 153.6$$
   - On Thursday, total items A sold = 360, percentage of items B = 56\%
     $$	ext{Items B on Thursday} = \frac{56}{100} 	imes 360 = 201.6$$

2. Find the average number of items B sold on Tuesday and Thursday:
   $$	ext{Average B} = \frac{153.6 + 201.6}{2} = \frac{355.2}{2} = 177.6$$

3. Calculate the average number of items A sold on Thursday and Friday:
   - Items A on Thursday = 360
   - Items A on Friday = 340
   $$	ext{Average A} = \frac{360 + 340}{2} = \frac{700}{2} = 350$$

4. Find the difference between these averages:
   $$350 - 177.6 = 172.4$$

**Note:** The original solution uses weighted averages based on total items sold, not just items A. Let's re-examine the problem carefully.

**Revised approach:**

- The table gives:
  - Items A sold each day
  - Percentage of items B and C sold each day

- Total items sold each day = Items A + Items B + Items C

- Since percentages of B and C are given, we can find total items sold by using the percentage of B or C with respect to total items.

Let's calculate total items sold on Tuesday and Thursday:

- On Tuesday:
  - Items A = 320
  - % of items B = 48\%
  - % of items C = 12\%

Since A + B + C = total,
  Let total items sold on Tuesday = T_tue

Items B = 48\% of T_tue
  Items C = 12\% of T_tue
  Items A = 320

So,
  $$320 + 0.48 T_{tue} + 0.12 T_{tue} = T_{tue}$$
  $$320 + 0.6 T_{tue} = T_{tue}$$
  $$320 = T_{tue} - 0.6 T_{tue} = 0.4 T_{tue}$$
  $$T_{tue} = \frac{320}{0.4} = 800$$

- Similarly, on Thursday:
  - Items A = 360
  - % of items B = 56\%
  - % of items C = 20\%

Let total items sold on Thursday = T_thu

$$360 + 0.56 T_{thu} + 0.20 T_{thu} = T_{thu}$$
  $$360 + 0.76 T_{thu} = T_{thu}$$
  $$360 = T_{thu} - 0.76 T_{thu} = 0.24 T_{thu}$$
  $$T_{thu} = \frac{360}{0.24} = 1500$$

- Now, calculate items B sold on Tuesday and Thursday:
  - Tuesday: $0.48 	imes 800 = 384$
  - Thursday: $0.56 	imes 1500 = 840$

- Average items B sold on Tuesday and Thursday:
  $$\frac{384 + 840}{2} = \frac{1224}{2} = 612$$

- Average items A sold on Thursday and Friday:
  - Thursday = 360
  - Friday = 340
  $$\frac{360 + 340}{2} = 350$$

- Required difference:
  $$612 - 350 = 262$$

**Answer:** 262

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