In a coaching class, some students register online, and some others register offline. No student registers both online and offline; hence the total registration number is the sum of online and offline registrations. The following facts and table pertain to these registration numbers for the five months – January to May of 2023. The table shows the minimum, maximum, median registration numbers of these five months, separately for online, offline and total number of registrations. The following additional facts are known.

1. In every month, both online and offline registration numbers were multiples of 10.
2. In January, the number of offline registrations was twice that of online registrations.
3. In April, the number of online registrations was twice that of offline registrations.
4. The number of online registrations in March was the same as the number of offline registrations in February.
5. The number of online registrations was the largest in May.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743775593/aptidude_questions/wkmjhzv0v1kexn3x9gr0.png)

---

Which pair of months definitely had the same total number of registrations?
I. January and April
II. February and May

Question

In a coaching class, some students register online, and some others register offline. No student registers both online and offline; hence the total registration number is the sum of online and offline registrations. The following facts and table pertain to these registration numbers for the five months – January to May of 2023. The table shows the minimum, maximum, median registration numbers of these five months, separately for online, offline and total number of registrations. The following additional facts are known.

1. In every month, both online and offline registration numbers were multiples of 10.
2. In January, the number of offline registrations was twice that of online registrations.
3. In April, the number of online registrations was twice that of offline registrations.
4. The number of online registrations in March was the same as the number of offline registrations in February.
5. The number of online registrations was the largest in May.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743775593/aptidude_questions/wkmjhzv0v1kexn3x9gr0.png)

---

Which pair of months definitely had the same total number of registrations?
I. January and April
II. February and May

Accepted Answer

**Correct Answer:**  
**D. Neither I nor II**

### Explanation:

To determine which pair of months definitely had the same total number of registrations, we analyze the given data and constraints step by step.

### Step-by-Step Analysis:

#### Key Observations from the Table
- **Online registrations (minimum = 40, maximum = 100, median = 80):**
  - The median indicates that **80** was one of the middle values for online registrations.
- **Offline registrations (minimum = 30, maximum = 80, median = 50):**
  - The median indicates that **50** was one of the middle values for offline registrations.
- **Total registrations (minimum = 110, maximum = 130, median = 120):**
  - The median indicates that **120** was one of the middle values for total registrations.

#### Additional Facts:
1. In January, **Offline = 2 × Online**.  
   - Total registrations in January: $$ 	ext{Total} = 	ext{Online} + 	ext{Offline} $$.
2. In April, **Online = 2 × Offline**.  
   - Total registrations in April: $$ 	ext{Total} = 	ext{Online} + 	ext{Offline} $$.
3. The number of online registrations in March was the same as the number of offline registrations in February.
4. May had the largest online registrations (**100**, as per maximum value).

### Verify Statements

#### Statement I: *January and April had the same total number of registrations.*
- From Fact 2 (January):
  - Let Online = **40**, Offline = **80**.
  - Total (January) = $$40 + 80 = 120$$.
- From Fact 3 (April):
  - Let Offline = **40**, Online = **80**.
  - Total (April) = $$40 + 80 = 120$$.

At first glance, both months seem to have the same total. However, this conclusion is not definite because:
- The table specifies only ranges and medians for total registrations, and other combinations could also satisfy these constraints.

Thus, Statement I cannot be concluded with certainty.

#### Statement II: *February and May had the same total number of registrations.*
- From Fact 4:
  - March’s Online = February’s Offline.
- February's total could be as low as:
$$
	ext{February Total} = 	ext{Online} + 	ext{Offline}.
$$
If Online = **60** and Offline = **50**, then February's Total = $$60 + 50 = 110$$.

- For May:
  - May had the largest online registration (**100**, as per Fact 5).
  - Offline must be at least **30** (minimum value for offline).
  - May's Total ≥ $$100 + 30 = 130$$.

Since February's total cannot match May's total under any valid configuration, Statement II is false.

### Conclusion:
Neither I nor II can be concluded with certainty, making option **D (Neither I nor II)** correct.

![Screenshot 2025-04-02 190658.png](https://res.cloudinary.com/aptidude/image/upload/v1743602748/aptidude_questions/peecpuddlx3m5eeooyro.png)![Screenshot 2025-04-02 190652.png](https://res.cloudinary.com/aptidude/image/upload/v1743602741/aptidude_questions/htdacmv2dhhn5n1wjsnw.png)![Screenshot 2025-04-02 190641.png](https://res.cloudinary.com/aptidude/image/upload/v1743602735/aptidude_questions/dmiwpfxl9qqmye0dt4sg.png)![Screenshot 2025-04-02 190634.png](https://res.cloudinary.com/aptidude/image/upload/v1743602727/aptidude_questions/ye3kpc3iltjhwiavyzvw.png)![Screenshot 2025-04-02 190625.png](https://res.cloudinary.com/aptidude/image/upload/v1743602707/aptidude_questions/ekul0kmrnhaytnvavvdm.png)![Screenshot 2025-04-02 190616.png](https://res.cloudinary.com/aptidude/image/upload/v1743602696/aptidude_questions/ziqst1lsbzirrwwlplzj.png)![Screenshot 2025-04-02 190610.png](https://res.cloudinary.com/aptidude/image/upload/v1743602688/aptidude_questions/wygea09k2j3hqhoi3f3c.png)

Answer

Answer

Answer