The ratio of two numbers is 5 : 6 and their LCM is 480, then their HCF is

Options

A. 69,15,22
B. 23,46,69,
C. 25,31,41
D. 23,21,35

Answer

To solve the problem, we need to find the HCF of two numbers given their ratio and LCM. Let's break down the steps:

### Step 1: Define the Numbers
Given the ratio of two numbers is 5:6, we can express the two numbers in terms of a variable \( x \):
- First number = \( 5x \)
- Second number = \( 6x \)

### Step 2: Use the LCM Formula
The relationship between the LCM (Least Common Multiple), HCF (Highest Common Factor), and the two numbers can be expressed as:
\[ \text{LCM} \times \text{HCF} = \text{First Number} \times \text{Second Number} \]
Substituting the values we have:
\[ 480 \times \text{HCF} = (5x) \times (6x) \]

### Step 3: Calculate the Product of the Numbers
Calculating the product of the two numbers:
\[ (5x) \times (6x) = 30x^2 \]
So, we can rewrite the equation:
\[ 480 \times \text{HCF} = 30x^2 \]

### Step 4: Express HCF in Terms of x
Now, we need to express HCF in terms of \( x \):
\[ \text{HCF} = \frac{30x^2}{480} \]
Simplifying this gives:
\[ \text{HCF} = \frac{x^2}{16} \]

### Step 5: Find the Value of x
To find \( x \), we need to determine the HCF. We know that the HCF must also be a common factor of both numbers \( 5x \) and \( 6x \). Since the ratio is 5:6, the HCF can be expressed as \( x \). Thus, we can set:
\[ \text{HCF} = x \]

### Step 6: Substitute Back to Find x
Now we can substitute \( x \) back into the equation:
\[ x = \frac{30x^2}{480} \]
Multiplying both sides by 480:
\[ 480x = 30x^2 \]
Rearranging gives:
\[ 30x^2 - 480x = 0 \]
Factoring out \( 30x \):
\[ 30x(x - 16) = 0 \]
This gives us two solutions: \( x = 0 \) or \( x = 16 \). Since \( x \) cannot be zero, we have:
\[ x = 16 \]

### Step 7: Find the HCF
Now substituting \( x \) back to find the HCF:
\[ \text{HCF} = x = 16 \]

### Final Answer
Thus, the HCF of the two numbers is **16**.

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