The mean score of 3 students in a test out of 25 is 18. Two new students take the test. What is the lowest marks that can be obtained by one newcomer who scores less than the other one for the overall average of the five students to rise to 20 ?

Options

A. 22
B. 21
C. 23
D. 20

Answer

To solve the problem, let's break it down step by step.

### Step 1: Calculate the total score of the first three students.
The mean score of the three students is 18. To find the total score of these students, we can use the formula for mean:

\[ \text{Mean} = \frac{\text{Total Score}}{\text{Number of Students}} \]

Given that the mean is 18 and the number of students is 3, we can rearrange the formula to find the total score:

\[ \text{Total Score} = \text{Mean} \times \text{Number of Students} = 18 \times 3 = 54 \]

### Step 2: Determine the desired total score for five students.
We want the average score of all five students to be 20. Therefore, we can calculate the total score needed for five students:

\[ \text{Total Score for 5 Students} = \text{Mean} \times \text{Number of Students} = 20 \times 5 = 100 \]

### Step 3: Calculate the total score needed from the two new students.
Now, we need to find out how much score the two new students must achieve together to reach the total score of 100:

\[ \text{Total Score from 2 New Students} = \text{Total Score for 5 Students} - \text{Total Score of 3 Students} = 100 - 54 = 46 \]

### Step 4: Set up the equation for the scores of the two new students.
Let’s denote the score of the first newcomer as \( x \) and the score of the second newcomer as \( y \). We know that \( x < y \) and:

\[ x + y = 46 \]

### Step 5: Find the lowest score for the first newcomer.
To find the lowest score for the first newcomer \( x \), we can express \( y \) in terms of \( x \):

\[ y = 46 - x \]

Since \( x < y \), we can substitute \( y \):

\[ x < 46 - x \]

Now, solving for \( x \):

\[ 2x < 46 \implies x < 23 \]

### Step 6: Determine the maximum possible value for \( x \).
Since \( x \) must be less than 23, the highest integer value \( x \) can take is 22. Therefore, if \( x = 22 \), we can find \( y \):

\[ y = 46 - 22 = 24 \]

### Conclusion:
The lowest score that can be obtained by the newcomer who scores less than the other one is:

\[ \boxed{22} \]