Finding Number of Students in a Class

Video Explanation

Question

Students are arranged in rows. If 3 students are extra in each row, there is 1 row less. If 3 students are less in each row, there are 2 rows more. Find the total number of students.

Solution

Step 1: Let Variables

Let number of students per row = \(x\)

Let number of rows = \(y\)

Total students = \(xy\)

—

Step 2: Form Equations

First condition:

\[ (x + 3)(y – 1) = xy \]

Expand:

\[ xy – x + 3y – 3 = xy \]

Cancel \(xy\):

\[ -x + 3y – 3 = 0 \]

\[ x = 3y – 3 \quad (1) \]

— Second condition:

\[ (x – 3)(y + 2) = xy \]

Expand:

\[ xy + 2x – 3y – 6 = xy \]

Cancel \(xy\):

\[ 2x – 3y – 6 = 0 \]

\[ 2x – 3y = 6 \quad (2) \]

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Step 3: Solve Linear Equations

Substitute (1) into (2):

\[ 2(3y – 3) – 3y = 6 \]

\[ 6y – 6 – 3y = 6 \]

\[ 3y = 12 \Rightarrow y = 4 \]

Then:

\[ x = 3(4) – 3 = 9 \]

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Step 4: Find Total Students

\[ \text{Total} = xy = 9 \times 4 = 36 \]

—

Conclusion

\[ \text{Total number of students} = 36 \]

Verification

(12 × 3 = 36) ✔

(6 × 6 = 36) ✔