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 total number of registrations in April?
Solution and Explanation
Given :
In every month,both online and offline registration numbers were multiples of 10.
From (2), in Jan, the number of offline registrations was double that of online registrations.
Let x be the number of online registrations
2x be the number of offline registrations
Total number of registrations be x + 2x = 3x
According to the table data :
3x should lie between the minimum and maximum total number of registrations.
x = 40 (as x should also be a multiple of 10)
In January, 40 and 80 are the online and offline registrations respectively.
From 5, we can say that the number of online registrations is highest in may.
In may, there are total 100 online registrations.
maximum possible total registrations is 130
lowest possible number of offline registrations is 30
Let x be the number of offline registrations in May which is equal to number of online registrations in March.
Let's arrange the data in table :
| Month | Online | Offline | Total |
|---|---|---|---|
| Jan | 40 | 80 | 120 |
| Feb | y | x | |
| Mar | x | z | |
| Apr | 80 | 140 | 120 |
| May | 100 | 30 | 130 |
From the table mentioned in question , 50 is the median for Offline data
Now x should lie between 50 and 80
For 80 to be the median for the online data
y should lie between 80 and 100
Now, let's consider the following :
Feb ⇒ Minimum value of y + x
= 80 + 50 = 130
Therefore , x = 50 and y = 80
Since,110 is the minimum number of total registrations, the only possibility is in March :
50 + z = 110
z = 60
Let's complete the table :
| Month | Online | Offline | Total |
|---|---|---|---|
| Jan | 40 | 80 | 120 |
| Feb | 80 | 50 | 130 |
| Mar | 50 | 60 | 130 |
| Apr | 80 | 140 | 120 |
| May | 100 | 30 | 130 |
So, total number of April's registrations is 120.
What was the number of online registrations in January?
Solution and Explanation
Given the conditions and data in the table, let’s analyze the situation for January:
1. From condition 2, in January, the number of offline registrations was twice that of online registrations.
2. Let the number of online registrations in January be \(x\). Then, the offline registrations would be \(2x\).
3. According to the table, the total registrations in January must fall within the minimum and maximum total values, which are 110 and 130, respectively.
4. Therefore, \( x + 2x = 3x \). We need \( 3x \) to be within the range 110 to 130. Since \( 3x = 120 \) (median total), we find \( x = 40 \).
5. Thus, the number of online registrations in January is \( 40 \).
Which of the following statements can be true?
I. The number of offline registrations was the smallest in May.
II. The total number of registrations was the smallest in February.
I. The number of offline registrations was the smallest in May.
II. The total number of registrations was the smallest in February.
- Only I
- Both I and II
- Neither I nor II
- Only II
The Correct Option is A
Solution and Explanation
To determine the correct answer, let’s analyze each statement based on the data provided:
1. Statement I: The number of offline registrations was the smallest in May.
- According to the table, the minimum number of offline registrations is 30. Given that May had the largest number of online registrations (from condition 5), it is reasonable to assume May could have the smallest offline registrations to keep the total consistent. Therefore, this statement can be true.
2. Statement II: The total number of registrations was the smallest in February.
- The total number of registrations ranges from 110 to 130, with 110 being the minimum. There is no specific information given that February had the smallest total, and this statement cannot be confirmed based on the table.
Hence, only Statement I can be true.
What best can be concluded about the number of offline registrations in February?
- 80
- 50
- 50 or 80
- 30 or 50 or 80
The Correct Option is B
Solution and Explanation
To determine the correct answer, let’s analyze the given options based on the data provided:
1. The table indicates that the median number of offline registrations is 50, meaning this value appears in the middle of the range of possible values for offline registrations over the months.
2. Given the conditions, and since no specific data contradicts this, we conclude that the most consistent value for offline registrations in February is 50, aligning with the median.
3. Therefore, the best conclusion about the number of offline registrations in February is 50.
Which pair of months definitely had the same total number of registrations?
I. January and April
II. February and May
I. January and April
II. February and May
- Only II
- Only 1
- Both I and II
- Neither I nor II
The Correct Option is C
Solution and Explanation
To determine the correct answer, let’s analyze each statement based on the conditions and the table data:
1. Statement I: January and April had the same total number of registrations.
- From the table, we know that the median total registrations are 120, which could likely apply to both January and April based on the given conditions. Thus, it is plausible that January and April have the same total number.
2. Statement II: February and May had the same total number of registrations.
- Similarly, based on the data and the conditions provided, it is plausible that February and May have the same total number of registrations.
Since both statements can be true, the correct answer is: Both I and II.
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