The ratio of two numbers is 4 : 5 and their LCM is 120. The numbers are

Options

A. 30,40
B. 40,32
C. 24,30
D. 36,20

Answer

To solve the problem, we will follow these steps:

1. **Understand the Given Information**: We have two numbers in the ratio of 4:5, and their Least Common Multiple (LCM) is 120.

2. **Express the Numbers in Terms of a Variable**: Let the two numbers be \( n_1 \) and \( n_2 \). Since their ratio is 4:5, we can express them as:
\[ n_1 = 4k \quad \text{and} \quad n_2 = 5k \]
where \( k \) is a common factor.

3. **Use the Relationship Between LCM, GCD, and the Numbers**: We know that the product of the LCM and the Greatest Common Divisor (GCD) of two numbers is equal to the product of the numbers themselves:
\[ \text{GCD} \times \text{LCM} = n_1 \times n_2 \]
Here, the GCD of \( n_1 \) and \( n_2 \) is \( k \), and the LCM is given as 120. Therefore, we can write:
\[ k \times 120 = (4k) \times (5k) \]

4. **Simplify the Equation**: Expanding the right side gives:
\[ k \times 120 = 20k^2 \]
Now, we can simplify this equation by dividing both sides by \( k \) (assuming \( k \neq 0 \)):
\[ 120 = 20k \]

5. **Solve for \( k \)**: To find \( k \), divide both sides by 20:
\[ k = \frac{120}{20} = 6 \]

6. **Find the Two Numbers**: Now that we have \( k \), we can find \( n_1 \) and \( n_2 \):
\[ n_1 = 4k = 4 \times 6 = 24 \]
\[ n_2 = 5k = 5 \times 6 = 30 \]

7. **Conclusion**: The two numbers are \( 24 \) and \( 30 \).

### Final Answer:
The two numbers are **24 and 30**.