The ratio of two numbers is 3 : 4 and their HCF is 5 . Their LCM is

Options

A. 10
B. 60
C. 15
D. 12

Answer

To find the LCM of two numbers given their ratio and HCF, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ratio**: The ratio of the two numbers is given as 3:4. This means we can represent the two numbers as:
- First number = 3x
- Second number = 4x
where x is a common multiplier.

2. **Identify the HCF**: The highest common factor (HCF) of the two numbers is given as 5.

3. **Express the Numbers in Terms of HCF**: Since the HCF is 5, we can express x in terms of the HCF:
- Since HCF = 5, we can set:
- 3x = 5a (for some integer a)
- 4x = 5b (for some integer b)
- From this, we can find that:
- x = 5/3a for the first number
- x = 5/4b for the second number

4. **Find the Value of x**: Since both expressions represent the same x, we can equate them:
- 3a = 4b
- This implies that a/b = 4/3. Therefore, we can take a = 4k and b = 3k for some integer k.

5. **Substituting Back to Find the Numbers**:
- Substitute a and b back into the expressions for the numbers:
- First number = 3x = 3(5/3)(4k) = 20k
- Second number = 4x = 4(5/4)(3k) = 15k

6. **Calculate the Product of the Numbers**: The product of the two numbers is:
- Product = (20k) * (15k) = 300k²

7. **Use the Relationship Between HCF, LCM, and Product**: The relationship between HCF, LCM, and the product of two numbers is given by:
- LCM * HCF = Product
- Let LCM = L, then:
- L * 5 = 300k²
- L = 300k² / 5
- L = 60k²

8. **Find the LCM**: Since k is a common factor and can be set to 1 for the simplest case, we find:
- LCM = 60 * 1² = 60

Thus, the LCM of the two numbers is **60**.