If the total of two numbers is 25 and its multiplication is 144, then their difference is

Options

A. 5
B. 7
C. 4
D. 6

Answer

To solve the problem step by step, we will follow the information given and derive the required difference between the two numbers.

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set Up the Equations
From the problem, we have two pieces of information:
1. The sum of the two numbers:
\[ x + y = 25 \quad \text{(Equation 1)} \]
2. The product of the two numbers:
\[ x \cdot y = 144 \quad \text{(Equation 2)} \]

### Step 3: Express One Variable in Terms of the Other
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 25 - x \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 3 into Equation 2
Now, substitute Equation 3 into Equation 2:
\[ x \cdot (25 - x) = 144 \]
Expanding this gives:
\[ 25x - x^2 = 144 \]

### Step 5: Rearrange the Equation
Rearranging the equation leads to:
\[ -x^2 + 25x - 144 = 0 \]
Multiplying through by -1 gives:
\[ x^2 - 25x + 144 = 0 \quad \text{(Equation 4)} \]

### Step 6: Factor the Quadratic Equation
To factor Equation 4, we need two numbers that multiply to 144 and add up to -25. The numbers are -9 and -16. Thus, we can write:
\[ (x - 9)(x - 16) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor equal to zero gives us the possible values for \( x \):
1. \( x - 9 = 0 \) → \( x = 9 \)
2. \( x - 16 = 0 \) → \( x = 16 \)

### Step 8: Find Corresponding Values of \( y \)
Using Equation 3, we can find the corresponding values of \( y \):
- If \( x = 9 \), then \( y = 25 - 9 = 16 \).
- If \( x = 16 \), then \( y = 25 - 16 = 9 \).

### Step 9: Calculate the Difference
Now, we can find the difference between the two numbers:
\[ |x - y| = |9 - 16| = 7 \quad \text{or} \quad |16 - 9| = 7 \]

### Final Answer
Thus, the difference between the two numbers is:
\[ \boxed{7} \]