The original average is the number of apples (300) divided by the number of students (let's call this "n").

Thus we begin with A₁ =  (300/n).

We add 10 students, which decreases the average by 1. But we still have the same number of apples.

Then the new average is A₂ =  (300/n+10)

We now can say A₁ - A₂  =  1

Substituting, we have:

       (300/n) - (300/n+10) = 1

Combining the two fractions on the left, we end up with

        (300n + 3000 - 300n)/n(n + 10)   =    1

          This reduces to 

              3000/n(n + 10) = 1

Simplifying, we get a quadratic equation:

         n² + 10n - 3000 = 0

Factoring, 

         (n - 50)(n + 60)  =  0

Solving, we find that n = 50 or n = -60.

The negative answer is meaningless because we can't have negative students, so we select n = 50.

       The original number of students was 50. Adding 10 made the number 60 and reduced the average number of apples from 6 to 5.

x = number of students

Since 300 apples are distributed among x students we will use the expression 300/x.

If there were 10 more students we would have x + 10, so we will use 300/(x + 10).  Also, since they would have recevied one less apple if there were 10 more students, we will call that 300/(x + 10) + 1.

Setting the two expression equl to each other, we have:

300/x = 300/(x + 10) + 1

If we multiply both sides by x(x + 10), we will have:

300(x + 10) = 300x + x(x + 10)

After distributing, we have

300x  + 3000 = 300x + x² + 10x

By subtracting everything on the right side, we have:

300x + 3000 - x² - 300x - 10x = 0

Simplified, we have

-x² -10x + 3000 = 0

After multiplying by -1, we have

x² + 10x - 3000 = 0

When we factor, we have:

(x - 50)(x + 60) = 0

Set each factor equal to zero and solve

x - 50 = 0        x + 60 = 0

x = 50            x = -60

Since you cannot have a negative number of students, the answer is 50 students.

 

Let A = number of apples each student received and S = number of students. we know that number of students times number of apples = 300, so A * S = 300 We are then told if there is a different number of students, there would be 1 less apple each. Therefore, (A - 1) * (S + 10) = 300 Solve the first equation for A, then substitute into the second equation and solve for S. Note that you will come up with two answers, one positive and one negative. One of these will be discarded because it won't make sense in the context of the problem.