The LCM of two numbers is 114 times their HCF. If the sum of LCM and HCF is 2300 and one of the numbers is 380, find the other number.

Options

A. 140
B. 160
C. 120
D. 190

Answer

To solve the problem step by step, we will use the relationship between the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two numbers.

### Step 1: Understand the relationship between LCM and HCF
We know that:
\[ \text{LCM} \times \text{HCF} = \text{Number 1} \times \text{Number 2} \]
Let’s denote:
- HCF as \( h \)
- LCM as \( l \)
- One of the numbers (given) as \( a = 380 \)
- The other number (unknown) as \( b \)

### Step 2: Set up the equations based on the problem statement
From the problem, we have:
1. \( l = 114h \) (LCM is 114 times HCF)
2. \( l + h = 2300 \) (The sum of LCM and HCF is 2300)

### Step 3: Substitute the LCM in the sum equation
Substituting \( l \) from the first equation into the second equation:
\[ 114h + h = 2300 \]
This simplifies to:
\[ 115h = 2300 \]

### Step 4: Solve for HCF
Now, divide both sides by 115 to find \( h \):
\[ h = \frac{2300}{115} = 20 \]

### Step 5: Find LCM using HCF
Now that we have \( h \), we can find \( l \):
\[ l = 114h = 114 \times 20 = 2280 \]

### Step 6: Use the relationship to find the other number
Using the relationship \( \text{LCM} \times \text{HCF} = \text{Number 1} \times \text{Number 2} \):
\[ 2280 \times 20 = 380 \times b \]
Calculating the left side:
\[ 45600 = 380 \times b \]

### Step 7: Solve for \( b \)
Now, divide both sides by 380 to find \( b \):
\[ b = \frac{45600}{380} = 120 \]

### Final Answer
The other number is \( b = 120 \).

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