The difference of two numbers is 1/9 of their sum. Their sum is 45. Find the LCM.

Options

A. 225
B. 100
C. 150
D. 200

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Set up the equations
Let the two numbers be \( A \) and \( B \). According to the problem, the difference of the two numbers is \( \frac{1}{9} \) of their sum. We can express this relationship mathematically as:
\[ A - B = \frac{1}{9}(A + B) \]

### Step 2: Use the given sum
We are also given that the sum of the two numbers is 45:
\[ A + B = 45 \]

### Step 3: Substitute the sum into the difference equation
Now, we can substitute \( A + B \) in the difference equation:
\[ A - B = \frac{1}{9}(45) \]
\[ A - B = 5 \]

### Step 4: Solve the system of equations
Now we have a system of two equations:
1. \( A + B = 45 \)
2. \( A - B = 5 \)

We can solve these equations simultaneously. Let's add both equations:
\[ (A + B) + (A - B) = 45 + 5 \]
This simplifies to:
\[ 2A = 50 \]
Now, divide both sides by 2:
\[ A = 25 \]

### Step 5: Find the value of B
Now that we have \( A \), we can substitute it back into one of the original equations to find \( B \):
\[ A + B = 45 \]
Substituting \( A = 25 \):
\[ 25 + B = 45 \]
Subtract 25 from both sides:
\[ B = 20 \]

### Step 6: Find the LCM of A and B
Now we need to find the LCM of \( A \) and \( B \), which are 25 and 20.

#### Step 6.1: Prime factorization
- The prime factorization of 25 is \( 5^2 \).
- The prime factorization of 20 is \( 2^2 \times 5^1 \).

#### Step 6.2: Calculate the LCM
To find the LCM, we take the highest power of each prime factor:
- For \( 2 \), the highest power is \( 2^2 \).
- For \( 5 \), the highest power is \( 5^2 \).

Thus, the LCM is:
\[ LCM = 2^2 \times 5^2 = 4 \times 25 = 100 \]

### Final Answer
The LCM of the two numbers is **100**.

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