If the sum of two numbers is 25 and their product is 156. Find the larger number.

Options

A. 11
B. 12
C. 13
D. 15

Answer

To solve the problem step by step, we can use the information given about the sum and product of two numbers. Let's denote the two numbers as \( A \) and \( B \).

### Step 1: Set up the equations
We know from the problem statement:
1. The sum of the two numbers:
\[ A + B = 25 \]
2. The product of the two numbers:
\[ A \times B = 156 \]

### Step 2: Express one variable in terms of the other
From the first equation, we can express \( B \) in terms of \( A \):
\[ B = 25 - A \]

### Step 3: Substitute into the product equation
Now, substitute \( B \) in the product equation:
\[ A \times (25 - A) = 156 \]

### Step 4: Expand and rearrange the equation
Expanding the left side gives:
\[ 25A - A^2 = 156 \]
Rearranging this gives:
\[ A^2 - 25A + 156 = 0 \]

### Step 5: Solve the quadratic equation
Now we can solve the quadratic equation using the quadratic formula:
\[ A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -25, c = 156 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-25)^2 - 4 \times 1 \times 156 = 625 - 624 = 1 \]

Now substituting back into the quadratic formula:
\[ A = \frac{25 \pm \sqrt{1}}{2 \times 1} = \frac{25 \pm 1}{2} \]

This gives us two possible values for \( A \):
\[ A = \frac{26}{2} = 13 \quad \text{and} \quad A = \frac{24}{2} = 12 \]

### Step 6: Determine the larger number
From the values we found:
- If \( A = 13 \), then \( B = 25 - 13 = 12 \).
- If \( A = 12 \), then \( B = 25 - 12 = 13 \).

Thus, the larger number is:
\[ \text{Larger number} = 13 \]

### Final Answer
The larger number is **13**.

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