Q 9 - Two numbers are in the ratio 8:11 . Considering their H.C.f as 6, find the numbers.

Given that the two numbers are in the ratio \(8:11\) and their Highest Common Factor (H.C.F.) is 6, we can determine the numbers using the following steps:

### Step 1: Express the numbers in terms of the H.C.F.
Let the two numbers be \(8x\) and \(11x\), where \(x\) is a common factor of both numbers.

### Step 2: Use the given H.C.F.
Since the H.C.F. of the two numbers is given as 6, and \(x\) is a common factor, we have:
\[
\text{H.C.F. of } 8x \text{ and } 11x = x \times \text{H.C.F. of } 8 \text{ and } 11
\]
Given that the H.C.F. of 8 and 11 is 1 (because 8 and 11 are coprime), the equation simplifies to:
\[
x \times 1 = 6
\]
Thus, \(x = 6\).

### Step 3: Find the numbers
Now that \(x = 6\), the two numbers are:
\[
8x = 8 \times 6 = 48
\]
\[
11x = 11 \times 6 = 66
\]

### Final Answer:
The two numbers are **48 and 66**.

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How do you determine if two numbers are coprime?
2. What is the difference between the H.C.F. and L.C.M. of two numbers?
3. If the ratio was different, say 3:4, how would the solution change?
4. Can two numbers with a given H.C.F. and a different ratio yield the same product?
5. How does the concept of prime factorization relate to finding the H.C.F.?

**Tip:** When dealing with ratios and H.C.F., always check if the given numbers in the ratio are coprime; this simplifies the problem significantly.

Two numbers are in the ratio 8:11. Considering their H.C.F as 6, find the numbers.

Learn how to find two numbers in the ratio 8:11 given their H.C.F. is 6. This problem involves key concepts like ratios, highest common factor (H.C.F.), and coprime numbers, suitable for students in Grades 6-8.

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