If 6 students occupy a bench, one bench will be left completely unoccupied. If 5 students occupy each bench, instead, one student will not find a seat. The number of students is

Question

Prepp Team · Accepted Answer

This problem involves setting up equations based on the given conditions about students and benches. We need to find the total number of students.

Students and Benches Setup

Let's use variables to represent the unknown quantities:

Let s be the total number of students.
Let b be the total number of benches.

Forming Equations from Conditions

We are given two scenarios, which will help us create two equations relating the number of students and benches.

Condition 1: 6 students per bench, one bench left unoccupied

If 6 students occupy each bench, and one bench is left completely empty, it means students are sitting only on the remaining benches. The number of occupied benches is the total number of benches minus the one empty bench, which is $\text{b} - 1$.

The total number of students is the number of occupied benches multiplied by the number of students per bench:

$s = 6 \times (b - 1)$

This gives us our first equation:

$s = 6b - 6 \quad (1)$

Condition 2: 5 students per bench, one student not finding a seat

If 5 students occupy each bench, and one student does not find a seat, it means all the benches are occupied by 5 students, and there is one extra student.

The total number of students is the number of benches multiplied by the number of students per bench, plus the one extra student:

$s = 5 \times b + 1$

This gives us our second equation:

$s = 5b + 1 \quad (2)$

Solving the Equations

Now we have a system of two linear equations with two variables (s and b):

$s = 6b - 6$

$s = 5b + 1$

Since both equations are equal to 's', we can set them equal to each other to solve for 'b':

$6b - 6 = 5b + 1$

Subtract $5b$ from both sides:

$6b - 5b - 6 = 1$

$b - 6 = 1$

Add 6 to both sides:

$b = 1 + 6$

$b = 7$

So, there are 7 benches.

Finding the Number of Students

Now that we know the number of benches ($b = 7$), we can substitute this value into either equation (1) or (2) to find the number of students (s).

Using equation (1):

$s = 6b - 6$

$s = 6(7) - 6$

$s = 42 - 6$

$s = 36$

Using equation (2):

$s = 5b + 1$

$s = 5(7) + 1$

$s = 35 + 1$

$s = 36$

Both equations give the same result. The number of students is 36.

Checking the Answer

Let's verify our answer with the given conditions:

If there are 36 students and 7 benches:
- 6 students per bench: $36 \div 6 = 6$ benches needed. Since there are 7 benches, $7 - 6 = 1$ bench is left empty. This matches Condition 1.
- 5 students per bench: $36 \div 5 = 7$ with a remainder of 1. This means 7 benches are filled with 5 students each, and 1 student is left over. This matches Condition 2.

The number of students is indeed 36.