7225 Students are sitting in an auditorium in such a way that there are as many students in a row as there are rows in the auditorium. How many students are there in a row in the auditorium?
85
Understanding the Auditorium Seating Problem
The problem describes a situation in an auditorium where the total number of students sitting is 7225. A key condition is given: the number of students in each row is equal to the number of rows in the auditorium.
Let's denote the number of rows by \(R\). Let's denote the number of students in each row by \(S\).
According to the problem statement, the number of students in a row is equal to the number of rows. So, we have:
\(S = R\)
The total number of students in the auditorium is the product of the number of rows and the number of students in each row. So, the total number of students is given by:
\(\text{Total Students} = \text{Number of Rows} \times \text{Students per Row}\)
Substituting the given total number of students and the condition \(S = R\), we get:
\(7225 = R \times S\)
Since \(S = R\), we can write:
\(7225 = R \times R\)
\(7225 = R^2\)
The question asks for the number of students in a row, which is \(S\). Since \(S = R\), we need to find the value of \(R\) that satisfies \(R^2 = 7225\). This means we need to calculate the square root of 7225.
\(R = \sqrt{7225}\)
And since \(S = R\), the number of students in a row is also \(\sqrt{7225}\).
Calculating the Number of Students per Row (\(\sqrt{7225}\))
To find the value of \(\sqrt{7225}\), we can look at the options provided or use methods like prime factorization or long division for square roots. A simpler method in a multiple-choice scenario is to square the options to see which one gives 7225.
| Option | Number of Students per Row | Square of the Number | Result |
|---|---|---|---|
| 1 | 75 | \(75^2\) | \(5625\) |
| 2 | 85 | \(85^2\) | \(7225\) |
| 3 | 95 | \(95^2\) | \(9025\) |
| 4 | 65 | \(65^2\) | \(4225\) |
Let's calculate the squares of the numbers given in the options:
- For option 1: \(75^2 = 75 \times 75 = 5625\)
- For option 2: \(85^2 = 85 \times 85 = 7225\)
- For option 3: \(95^2 = 95 \times 95 = 9025\)
- For option 4: \(65^2 = 65 \times 65 = 4225\)
Comparing the calculated squares with the total number of students (7225), we see that \(85^2\) equals 7225.
Therefore, the number of students in a row is 85.
This also means the number of rows is 85.
So, there are 85 rows, and each row has 85 students. The total number of students is \(85 \times 85 = 7225\), which matches the given information.
Conclusion
Based on the calculation, the number of students in a row in the auditorium is 85.
Important Questions from Linear Equation in 1 Variable
What should be subtracted from p and added to q so that the resulting ratio becomes 1 : 5?
The cost of a pen is five times the cost of a pencil. I bought 8 pens and 4 pencils for Rs. 132. Find the cost of 5 pens and 5 pencils.
Find the value of k, for which the system of equations kx + 3y = 26 and 21x + (k + 2)y = 71 + k has infinitely many solutions.
If 2x 2 + 5x + 1 = 0, then one of the values of \(\frac{1}{2}\left( {x\, - \,\frac{1}{{2x}}} \right)\) is:
Solve: 5(x + 43) = 2(3x + 4)
