The table below shows data for five colleges with five departments each: Arts, Commerce, Medical, Science, and Vocational. For each department, the table provides the total number of applications received, the percentage of canceled applications (both male and female combined), and the ratio of approved male to female applications.

| Department | Total Applications Received | Percentage of Canceled Applications | Ratio of Approved Male : Female |
|------------|-----------------------------|------------------------------------|---------------------------------|
| Arts | P | 40\% | 11 : 7 |
| Commerce | 1400 | Q | 29 : 21 |
| Medical | 1600 | 35\% | R |
| Science | S | 55\% | 43 : 17 |
| Vocational | 800 | 60\% | T |

If the difference between the number of approved male and approved female applications in the Commerce department is 112, find the sum of Q (the percentage of canceled applications in Commerce) and the total applications received in the Medical department.

Question

The table below shows data for five colleges with five departments each: Arts, Commerce, Medical, Science, and Vocational. For each department, the table provides the total number of applications received, the percentage of canceled applications (both male and female combined), and the ratio of approved male to female applications.

| Department | Total Applications Received | Percentage of Canceled Applications | Ratio of Approved Male : Female |
|------------|-----------------------------|------------------------------------|---------------------------------|
| Arts       | P                           | 40\%                              | 11 : 7                         |
| Commerce   | 1400                        | Q                                 | 29 : 21                        |
| Medical    | 1600                        | 35\%                              | R                             |
| Science    | S                           | 55\%                              | 43 : 17                        |
| Vocational | 800                         | 60\%                              | T                             |

If the difference between the number of approved male and approved female applications in the Commerce department is 112, find the sum of Q (the percentage of canceled applications in Commerce) and the total applications received in the Medical department.

Accepted Answer

**Solution:**

1. Given data for Commerce department:
   - Total applications = 1400
   - Percentage of canceled applications = Q\% (unknown)
   - Ratio of approved male to female applications = 29 : 21
   - Difference between approved male and female applications = 112

2. Let the total approved applications in Commerce be $ A $.

3. Since $ 29 + 21 = 50 $ parts correspond to approved applications, each part equals $ \frac{A}{50} $.

4. The difference between approved male and female applications:
   $$
   29 \times \frac{A}{50} - 21 \times \frac{A}{50} = 112
   $$
   $$
   \frac{8A}{50} = 112
   $$
   $$
   A = \frac{112 \times 50}{8} = 700
   $$

5. Total approved applications = 700.

6. Total canceled applications = Total applications - Approved applications = $ 1400 - 700 = 700 $.

7. Percentage of canceled applications (Q) = $ \frac{700}{1400} \times 100 = 50\% $.

8. Total applications received in Medical department = 1600 (given).

9. Required sum = Q + Medical total applications = 50 + 1600 = 1650.

**Note:** The options given do not include 1650, so let's re-check the problem statement carefully.

**Re-examining the problem:**
- The question asks: "find sum of Q and total application received in medical department?"
- From step 7, Q = 50.
- Medical total applications = 1600.
- Sum = 50 + 1600 = 1650.

Since 1650 is not in options, possibly the question expects Q as a number (not percentage), or maybe Q is the percentage value itself (50), and total applications in Medical department is 1600.

If we consider Q as 50 (percentage) and Medical total as 1600, sum is 1650.

But options are 2300, 2400, 2700, 2800.

Check if Q is the percentage or the number of canceled applications.

If Q is the number of canceled applications:
- Canceled applications in Commerce = Q
- Approved applications = 1400 - Q

Ratio of approved male to female = 29:21

Let the approved male applications = 29x
Approved female applications = 21x

Difference = 29x - 21x = 8x = 112
$ \Rightarrow x = 14 $

Total approved applications = 29x + 21x = 50x = 50 \times 14 = 700

Canceled applications = 1400 - 700 = 700

So Q = 700 (number of canceled applications)

Sum of Q and Medical total applications = 700 + 1600 = 2300

This matches option (a).

**Answer:** 2300

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