The ratio of two numbers is 4 : 5 and their H.C.F is 4. Find their L.C.M.

Options

A. a. 96
B. b. 80
C. c. 73
D. d. 48

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Understand the Ratio
The ratio of the two numbers is given as 4:5. This means that if we let the two numbers be represented as \( 4x \) and \( 5x \), where \( x \) is a common multiplier.

### Step 2: Express the Numbers
Let the two numbers be:
- First number = \( 4x \)
- Second number = \( 5x \)

### Step 3: Use the H.C.F
We are given that the H.C.F (Highest Common Factor) of these two numbers is 4. Since the H.C.F of \( 4x \) and \( 5x \) is \( x \), we can equate:
\[ x = 4 \]

### Step 4: Calculate the Actual Numbers
Now, substituting \( x = 4 \) into the expressions for the two numbers:
- First number = \( 4x = 4 \times 4 = 16 \)
- Second number = \( 5x = 5 \times 4 = 20 \)

### Step 5: Find the L.C.M
We can use the relationship between L.C.M (Least Common Multiple), H.C.F, and the two numbers:
\[ \text{L.C.M} \times \text{H.C.F} = \text{First number} \times \text{Second number} \]

Substituting the values we have:
- H.C.F = 4
- First number = 16
- Second number = 20

So,
\[ \text{L.C.M} \times 4 = 16 \times 20 \]

### Step 6: Calculate the Product
Now, calculate \( 16 \times 20 \):
\[ 16 \times 20 = 320 \]

### Step 7: Solve for L.C.M
Now, substituting back into the equation:
\[ \text{L.C.M} \times 4 = 320 \]

To find the L.C.M, divide both sides by 4:
\[ \text{L.C.M} = \frac{320}{4} = 80 \]

### Final Answer
Thus, the L.C.M of the two numbers is **80**.

---