The product of two numbers is 24 times the difference of these two numbers. If the sum of these numbers is 14, the larger number is

Options

A. 9
B. 8
C. 7
D. 10

Answer

To solve the problem step by step, we need to find two numbers based on the given conditions. Let's denote the two numbers as \( x \) and \( y \).

### Step 1: Set up the equations
From the problem, we have two key pieces of information:
1. The product of the two numbers is 24 times their difference.
2. The sum of the two numbers is 14.

We can express these conditions mathematically:
1. \( xy = 24(x - y) \)
2. \( x + y = 14 \)

### Step 2: Express one variable in terms of the other
From the second equation, we can express \( y \) in terms of \( x \):
\[ y = 14 - x \]

### Step 3: Substitute into the first equation
Now, we substitute \( y \) in the first equation:
\[ x(14 - x) = 24(x - (14 - x)) \]

### Step 4: Simplify the equation
Now, simplify both sides:
\[ 14x - x^2 = 24(2x - 14) \]
\[ 14x - x^2 = 48x - 336 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ -x^2 + 14x - 48x + 336 = 0 \]
\[ -x^2 - 34x + 336 = 0 \]
Multiplying through by -1:
\[ x^2 + 34x - 336 = 0 \]

### Step 6: Factor the quadratic equation
Now we need to factor the quadratic equation. We look for two numbers that multiply to \(-336\) and add to \(34\). The factors are \(42\) and \(-8\):
\[ (x + 42)(x - 8) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
\[ x + 42 = 0 \quad \Rightarrow \quad x = -42 \quad (\text{not valid since we are looking for positive numbers}) \]
\[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \]

### Step 8: Find \( y \)
Now, substitute \( x = 8 \) back into the equation for \( y \):
\[ y = 14 - x = 14 - 8 = 6 \]

### Step 9: Identify the larger number
The larger number between \( x \) and \( y \) is:
\[ \text{Larger number} = \max(8, 6) = 8 \]

### Final Answer
The larger number is **8**.