Comprehension:

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.

	Minimum	Maximum	Median
Online	40	100	80
Offline	30	80	50
Total	110	130	120

What was the number of online registrations in January?

To find the number of online registrations in January, we can analyze the given conditions and the table showing the minimum, maximum, and median registration numbers.

Let us denote the online and offline registrations in January as ${\mathrm{ON}}_{\mathrm{Jan}}$ and ${\mathrm{OF}}_{\mathrm{Jan}}$ respectively.

From the given table, we have the following parameters for online and offline registrations:

• Online Registrations: Minimum = 40, Maximum = 100, Median = 80
• Offline Registrations: Minimum = 30, Maximum = 80, Median = 50

From Fact 2, we know that in January, the number of offline registrations was twice that of online registrations:

${\mathrm{OF}}_{\mathrm{Jan}}=2·{\mathrm{ON}}_{\mathrm{Jan}}$

Since the minimum possible online registration in any month is 40, we must have:

${\mathrm{ON}}_{\mathrm{Jan}}\ge 40$

Substituting this minimum value into the relationship for January registrations, we get:

${\mathrm{OF}}_{\mathrm{Jan}}\ge 2·40=80$

However, from the table, the maximum possible offline registration in any month is 80. Therefore, the offline registrations in January cannot exceed 80:

${\mathrm{OF}}_{\mathrm{Jan}}\le 80$

Comparing the two inequalities, the only possible value for the offline registrations in January is:

${\mathrm{OF}}_{\mathrm{Jan}}=80$

Using the relationship ${\mathrm{OF}}_{\mathrm{Jan}}=2·{\mathrm{ON}}_{\mathrm{Jan}}$, we can solve for the online registrations in January:

${\mathrm{ON}}_{\mathrm{Jan}}=\frac{{\mathrm{OF}}_{\mathrm{Jan}}}{2}=\frac{80}{2}=40$

Thus, the number of online registrations in January was 40.