Two numbers are such that their difference, their sum and their product are in the ratio of 1 : 7 : 24. The product of the numbers is

Options

A. 24
B. 36
C. 48
D. 60

Answer

To solve the problem, we need to find two numbers whose difference, sum, and product are in the ratio of 1:7:24. Let's denote the two numbers as \( A \) and \( B \).

### Step-by-Step Solution:

1. **Define the Ratios**:
- Let the difference \( A - B = x \) (where \( x \) is a common multiplier).
- Let the sum \( A + B = 7x \).
- Let the product \( A \times B = 24x \).

2. **Set Up the Equations**:
- From the definitions, we have:
\[ A - B = x \quad (1) \]
\[ A + B = 7x \quad (2) \]

3. **Add Equations (1) and (2)**:
- Adding these two equations gives:
\[ (A - B) + (A + B) = x + 7x \]
\[ 2A = 8x \]
- Therefore, we can solve for \( A \):
\[ A = 4x \quad (3) \]

4. **Subtract Equation (1) from Equation (2)**:
- Subtracting equation (1) from equation (2) gives:
\[ (A + B) - (A - B) = 7x - x \]
\[ 2B = 6x \]
- Therefore, we can solve for \( B \):
\[ B = 3x \quad (4) \]

5. **Find the Product**:
- Now we can substitute \( A \) and \( B \) into the product equation:
\[ A \times B = (4x) \times (3x) = 12x^2 \]
- We know from the product ratio that:
\[ A \times B = 24x \]
- Setting these equal gives:
\[ 12x^2 = 24x \]

6. **Solve for \( x \)**:
- Rearranging gives:
\[ 12x^2 - 24x = 0 \]
- Factoring out \( 12x \):
\[ 12x(x - 2) = 0 \]
- This gives us two solutions: \( x = 0 \) or \( x = 2 \). Since \( x = 0 \) does not make sense in this context, we take \( x = 2 \).

7. **Calculate \( A \) and \( B \)**:
- Substitute \( x = 2 \) back into equations (3) and (4):
\[ A = 4x = 4 \times 2 = 8 \]
\[ B = 3x = 3 \times 2 = 6 \]

8. **Final Product**:
- Now we can find the product:
\[ A \times B = 8 \times 6 = 48 \]

### Conclusion:
The product of the two numbers is **48**.