300 apples are distributed equally among a certain number of students. Had three been 10 more students, each would have received one apple less. Find the number of students.

Explanation

To solve the problem, we need to find the number of students (let's denote it as \( n \)) who are sharing 300 apples. The problem states that if there were 10 more students, each student would receive one apple less.

### Step 1: Set up the equation
Let \( n \) be the number of students. The number of apples each student receives is given by:
\[ \text{Apples per student} = \frac{300}{n} \]
If there were 10 more students, the number of students would be \( n + 10 \), and the number of apples each would receive is:
\[ \text{Apples per student with 10 more students} = \frac{300}{n + 10} \]
According to the problem, if there were 10 more students, each student would receive one apple less. Therefore, we can set up the equation:
\[ \frac{300}{n} - 1 = \frac{300}{n + 10} \]

### Step 2: Clear the fractions
To eliminate the fractions, we can multiply both sides of the equation by \( n(n + 10) \):
\[ 300(n + 10) - n(n + 10) = 300n \]

### Step 3: Expand and simplify the equation
Expanding both sides gives us:
\[ 300n + 3000 - n^2 - 10n = 300n \]
Now, we can simplify this:
\[ 3000 - n^2 - 10n = 0 \]

### Step 4: Rearrange the equation
Rearranging the equation gives us:
\[ n^2 + 10n - 3000 = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula to solve for \( n \):
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 10 \), and \( c = -3000 \). Plugging in these values:
\[ n = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot (-3000)}}{2 \cdot 1} \]
\[ n = \frac{-10 \pm \sqrt{100 + 12000}}{2} \]
\[ n = \frac{-10 \pm \sqrt{12100}}{2} \]
\[ n = \frac{-10 \pm 110}{2} \]

### Step 6: Calculate the possible values for \( n \)
Calculating the two possible values:
1. \( n = \frac{100}{2} = 50 \)
2. \( n = \frac{-120}{2} = -60 \) (not a valid solution since the number of students cannot be negative)

Thus, the number of students is:
\[ n = 50 \]

### Final Answer
The number of students is **50**.

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