Question

The students of a class are made to stand in rows (complete). If one student is extra in a row, there would be 2 rows less and if one student is less in a row, there would be 3 rows more. Find the number of students in the class.

Answer 1

Given:

• Students are arranged in rows with an unknown number of students per row and total rows.
• If one student is added to each row, the number of rows decreases by 2.
• If one student is removed from each row, the number of rows increases by 3.
• We need to find the total number of students in the class.

Step-wise calculation:

1. Let the original number of students per row be (x) and the original number of rows be (y).

Then the total number of students (N = x × y).

2. When there is one extra student in each row, the number of students per row becomes (x + 1) and the number of rows decreases by 2, so the rows become (y – 2).

The total number of students remains the same, so:

(x + 1)(y – 2) = N

(x + 1)(y – 2) = xy

3. When there is one less student in each row, the number of students per row is (x – 1) and the number of rows increases by 3, so rows become (y + 3).

Again, total students remain:

(x – 1)(y + 3) = N

(x – 1)(y + 3) = xy

From equation 2:

xy = (x + 1)(y – 2)

xy = xy – 2x + y – 2 

Simplify:

xy = xy – 2x + y – 2

0 = –2x + y – 2 

2x = y – 2

⇒ y = 2x + 2

From equation 3:

xy = (x – 1)(y + 3)

xy = xy + 3x – y – 3

Simplify:

xy = xy + 3x – y – 3 

0 = 3x – y – 3 

y = 3x – 3

Set the two expressions for (y) equal:

2x + 2 = 3x – 3 

3 + 2 = 3x – 2x 

5 = x

Now substitute (x = 5) into (y = 2x + 2):

y = 2(5) + 2

y = 10 + 2

y = 12

Number of students

N = x × y

N = 5 × 12

N = 60

The number of students in the class is 60.