Ratio and LCM

Given:
- Two numbers are in the ratio 4:5.
- The LCM (Least Common Multiple) of the two numbers is 180.

To find:
- The smaller number.

Solution:

First, let's understand what ratio and LCM mean.

Ratio:
- A ratio represents the proportional relationship between two or more numbers.
- It is expressed in the form of a:b, where a and b are the numbers in the ratio.
- In this case, the ratio is given as 4:5.

LCM (Least Common Multiple):
- LCM is the smallest multiple that is common to two or more numbers.
- In this case, the LCM of the two numbers is given as 180.

Step 1: Identify the common factor
- To find the smaller number, we need to determine the common factor between the given ratio (4:5) and the LCM (180).
- Since the LCM represents the smallest multiple, it should be a multiple of both numbers in the ratio.

Step 2: Find the common multiple
- We can determine the common multiple by multiplying the two numbers in the ratio by the same factor.
- In this case, we need to find a factor that will make both 4 and 5 multiples of it.
- Let's start by trying the factors of 4: 1, 2, 4. None of them is a multiple of 5.
- Now, let's try the factors of 5: 1, 5. The factor 5 is a multiple of 4 (5 * 4 = 20).

Step 3: Calculate the smaller number
- To find the smaller number, we need to divide the common multiple (20) by the factor we used (5).
- 20 ÷ 5 = 4.
- Therefore, the smaller number is 4.

Step 4: Check the answer
- To confirm our answer, we can calculate the larger number using the ratio.
- The larger number is found by multiplying the smaller number by the ratio factor: 4 * 5 = 20.
- The LCM of the two numbers (4 and 20) is 20, which matches the given LCM (180).

Conclusion:
The smaller number in the ratio 4:5, with an LCM of 180, is 4. Therefore, the correct answer is option 'C' (36).