A certain number of tennis balls were purchased for 450. Five more balls could have been purchased for the same amount if each ball was cheaper by 15. Find the number of balls purchased.

Options

A. 15
B. 20
C. 10
D. 25

Answer

To solve the problem step by step, let's denote the number of tennis balls purchased as \( x \) and the price of each tennis ball as \( p \).

### Step 1: Set up the equation for the total cost
The total cost for purchasing \( x \) tennis balls is given by:
\[ xp = 450 \]

### Step 2: Set up the equation for the alternative scenario
If each ball was cheaper by 15, the new price per ball would be \( p - 15 \). According to the problem, if the price was reduced by 15, then 5 more balls could be purchased for the same total amount of 450. Thus, the equation for this scenario is:
\[ (x + 5)(p - 15) = 450 \]

### Step 3: Expand the second equation
Expanding the second equation gives:
\[ xp - 15x + 5p - 75 = 450 \]

### Step 4: Substitute the value of \( xp \)
From Step 1, we know that \( xp = 450 \). Substituting this into the expanded equation:
\[ 450 - 15x + 5p - 75 = 450 \]
This simplifies to:
\[ -15x + 5p - 75 = 0 \]
Rearranging gives:
\[ 5p = 15x + 75 \]
Dividing through by 5:
\[ p = 3x + 15 \]

### Step 5: Substitute \( p \) back into the first equation
Now we substitute \( p \) back into the first equation \( xp = 450 \):
\[ x(3x + 15) = 450 \]
Expanding this gives:
\[ 3x^2 + 15x = 450 \]

### Step 6: Rearranging the equation
Rearranging the equation to set it to zero:
\[ 3x^2 + 15x - 450 = 0 \]
Dividing the entire equation by 3 simplifies it:
\[ x^2 + 5x - 150 = 0 \]

### Step 7: Factor the quadratic equation
Now we need to factor the quadratic equation:
\[ (x + 15)(x - 10) = 0 \]
Setting each factor to zero gives:
\[ x + 15 = 0 \quad \text{or} \quad x - 10 = 0 \]

### Step 8: Solve for \( x \)
From \( x + 15 = 0 \), we get \( x = -15 \) (not valid since the number of balls cannot be negative).
From \( x - 10 = 0 \), we get \( x = 10 \).

### Conclusion
Thus, the number of tennis balls purchased is:
\[ \boxed{10} \]