In a classroom, if 6 students per bench are assigned to accommodate all students, one more bench will be required. However, if 7 students are accommodated per bench, there would be space left for 5 students. What in the number of student in the class?

Options

A. a)30
B. b)42
C. c)72
D. d)none of these

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given in the question.

### Step 1: Define the variables
Let:
- \( x \) = total number of students in the class
- \( y \) = number of benches used

### Step 2: Set up the first equation
According to the problem, if 6 students are assigned to each bench, one more bench will be required. This can be expressed mathematically as:
\[ \frac{x}{6} = y + 1 \]
This means that the total number of students divided by 6 gives the number of benches used plus one additional bench.

### Step 3: Rearranging the first equation
Rearranging the first equation gives us:
\[ x = 6(y + 1) \quad \Rightarrow \quad x = 6y + 6 \]

### Step 4: Set up the second equation
The problem also states that if 7 students are accommodated per bench, there would be space left for 5 students. This can be expressed as:
\[ \frac{x}{7} = y - \frac{5}{7} \]
This means that the total number of students divided by 7 gives the number of benches used, minus the space for 5 students.

### Step 5: Rearranging the second equation
Rearranging the second equation gives us:
\[ x = 7y - 5 \]

### Step 6: Equate the two expressions for \( x \)
Now we have two expressions for \( x \):
1. \( x = 6y + 6 \)
2. \( x = 7y - 5 \)

Setting these two equations equal to each other:
\[ 6y + 6 = 7y - 5 \]

### Step 7: Solve for \( y \)
Rearranging gives:
\[ 6 + 5 = 7y - 6y \]
\[ 11 = y \]

### Step 8: Substitute \( y \) back to find \( x \)
Now that we have \( y \), substitute it back into one of the equations to find \( x \). Using \( x = 6y + 6 \):
\[ x = 6(11) + 6 = 66 + 6 = 72 \]

### Conclusion
The total number of students in the class is \( x = 72 \).