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

Which pair of months definitely had the same total number of registrations?

I. January and April

II. February and May

To determine which pairs of months had the same total number of registrations, we can analyze the given conditions and limits step-by-step.

Let the online, offline, and total registrations for month M be denoted as ${\text{On}}_M$, ${\text{Off}}_M$, and ${\text{Tot}}_M$ respectively. We are given that ${\text{Tot}}_M={\text{On}}_M+{\text{Off}}_M$, and all registration numbers are multiples of 10.

From the given table:
- Online registrations ($\text{On}$): Minimum = 40, Maximum = 100, Median = 80.
- Offline registrations ($\text{Off}$): Minimum = 30, Maximum = 80, Median = 50.
- Total registrations ($\text{Tot}$): Minimum = 110, Maximum = 130, Median = 120. Since they are multiples of 10, the only possible total registrations for any month are 110, 120, or 130.

Step 1: Analyze January (Jan) and April (Apr)
- Fact 2: In January, the offline registrations were twice that of online registrations:

${\text{Off}}_{\text{Jan}}=2\times {\text{On}}_{\text{Jan}}$

Since ${\text{On}}_{\text{Jan}}\ge 40$ (the minimum online value), we must have ${\text{Off}}_{\text{Jan}}\ge 80$. Since the maximum offline value is 80, we get:

${\text{On}}_{\text{Jan}}=40$, ${\text{Off}}_{\text{Jan}}=80$

Thus, the total registrations in January is:

${\text{Tot}}_{\text{Jan}}=40+80=120$

- Fact 3: In April, the online registrations were twice that of offline registrations:

${\text{On}}_{\text{Apr}}=2\times {\text{Off}}_{\text{Apr}}$

If ${\text{Off}}_{\text{Apr}}=30$, then ${\text{On}}_{\text{Apr}}=60$, giving a total of 90 (which is less than the minimum total of 110).
If ${\text{Off}}_{\text{Apr}}=40$, then ${\text{On}}_{\text{Apr}}=80$, giving a total of 120.
If ${\text{Off}}_{\text{Apr}}\ge 50$, then ${\text{On}}_{\text{Apr}}\ge 100$, giving a total of at least 150 (which exceeds the maximum total of 130).
Therefore, we must have:

${\text{Off}}_{\text{Apr}}=40$, ${\text{On}}_{\text{Apr}}=80$

Thus, the total registrations in April is:

${\text{Tot}}_{\text{Apr}}=80+40=120$

Since both January and April have a total of 120, Statement I is correct.

Step 2: Analyze February (Feb) and May (May)
- Fact 5: Online registrations were the largest in May. Therefore:

${\text{On}}_{\text{May}}=100$ (the maximum online value)

The total registrations for May is ${\text{Tot}}_{\text{May}}=100+{\text{Off}}_{\text{May}}$. Since the minimum for $\text{Off}$ is 30, and the maximum for $\text{Tot}$ is 130, we must have:

${\text{Off}}_{\text{May}}=30$, which gives ${\text{Tot}}_{\text{May}}=100+30=130$

- Fact 4: ${\text{On}}_{\text{March}}={\text{Off}}_{\text{Feb}}$.
Let us find the possible values. Since the sorted offline values are 30 (May), 40 (April), 50 (median), ${\text{Off}}_{\text{Feb}}$, and 80 (January), we know that ${\text{Off}}_{\text{Feb}}$ must be at least 50.
If ${\text{Off}}_{\text{Feb}}\ge 50$, then ${\text{On}}_{\text{March}}\ge 50$.
Since ${\text{On}}_{\text{Apr}}=80$ is the median of online registrations, the lower two online registration values are for Jan (40) and either Feb or March.
If Feb has the lower online registration, then ${\text{On}}_{\text{Feb}}\le 80$, but since 80 is the median and 100 is the max, the remaining online value (for Feb) must be at least 80.
Thus, ${\text{On}}_{\text{Feb}}\ge 80$.
With ${\text{On}}_{\text{Feb}}\ge 80$ and ${\text{Off}}_{\text{Feb}}\ge 50$, the total is:

${\text{Tot}}_{\text{Feb}}={\text{On}}_{\text{Feb}}+{\text{Off}}_{\text{Feb}}\ge 80+50=130$

Since the maximum possible total registration is 130, we must have:

${\text{Tot}}_{\text{Feb}}=130$

Since both February and May have a total of 130, Statement II is correct.

Thus, both pairs of months definitely had the same total number of registrations.