The ratio of two numbers is 15:11 . If their HCF be 13 then these numbers will be:

Options

A. `15:11`
B. `75:55`
C. `105:77`
D. `195:143`

Answer

To solve the problem, we need to find two numbers that are in the ratio of 15:11 and have a highest common factor (HCF) of 13. Here's how we can do this step by step:

### Step 1: Understand the Ratio
The ratio of the two numbers is given as 15:11. This means that if we let the first number be \( 15x \) and the second number be \( 11x \), where \( x \) is a common multiplier.

### Step 2: Use the HCF
We are also given that the HCF of these two numbers is 13. This implies that both numbers can be expressed as multiples of 13. Therefore, we can express the numbers in terms of 13.

### Step 3: Set Up the Equations
Since the HCF is 13, we can write:
- First number = \( 15x = 13a \)
- Second number = \( 11x = 13b \)

Where \( a \) and \( b \) are integers.

### Step 4: Find the Value of \( x \)
From the ratio \( 15:11 \), we can express \( x \) in terms of 13:
- Let \( x = 13 \) (since the HCF is 13).

### Step 5: Calculate the Numbers
Now we can substitute \( x \) back into the expressions for the numbers:
- First number = \( 15 \times 13 = 195 \)
- Second number = \( 11 \times 13 = 143 \)

### Step 6: Final Answer
Thus, the two numbers are 195 and 143.

### Summary
The two numbers that are in the ratio of 15:11 and have an HCF of 13 are:
- **First Number: 195**
- **Second Number: 143**

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