The ratio of the sum to the LCM of two natural numbers is 7 : 12 . If their HCF is 4, then the smaller numbers is

Options

A. 20
B. 16
C. 12
D. 8

Answer

To solve the problem step by step, we will follow the given information and derive the smaller number based on the ratio of the sum to the LCM of two natural numbers and their HCF.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- The ratio of the sum of two natural numbers to their LCM is given as \(7:12\).
- The HCF of the two numbers is \(4\).

2. **Let the Two Numbers be:**
- Since the HCF is \(4\), we can express the two numbers as:
\[ a = 4x \quad \text{and} \quad b = 4y \]
- Here, \(x\) and \(y\) are coprime integers (since their HCF is 1).

3. **Calculate the Sum and LCM:**
- The sum of the two numbers:
\[ a + b = 4x + 4y = 4(x + y) \]
- The LCM of the two numbers:
\[ \text{LCM}(a, b) = \frac{a \cdot b}{\text{HCF}(a, b)} = \frac{(4x)(4y)}{4} = 4xy \]

4. **Set Up the Ratio:**
- According to the problem, the ratio of the sum to the LCM is:
\[ \frac{4(x + y)}{4xy} = \frac{x + y}{xy} \]
- This ratio is given as \( \frac{7}{12} \):
\[ \frac{x + y}{xy} = \frac{7}{12} \]

5. **Cross Multiplying:**
- Cross-multiply to eliminate the fraction:
\[ 12(x + y) = 7xy \]

6. **Rearranging the Equation:**
- Rearranging gives:
\[ 7xy - 12x - 12y = 0 \]

7. **Using the Quadratic Formula:**
- We can rewrite this as:
\[ 7xy - 12x - 12y + 144 = 144 \]
- This can be factored or solved using the quadratic formula. However, we will look for integer solutions for \(x\) and \(y\).

8. **Finding Integer Solutions:**
- We can try possible values for \(x\) and \(y\) that satisfy \(x + y = 7\) and \(xy = 12\).
- The pairs \((3, 4)\) and \((4, 3)\) satisfy these equations.

9. **Calculating the Smaller Number:**
- Since \(x = 3\) and \(y = 4\) (or vice versa), we calculate the smaller number:
\[ \text{Smaller Number} = 4x = 4 \times 3 = 12 \]

### Final Answer:
The smaller number is **12**.