Three numbers are in the ratio 1 : 2 : 3 and their HCF is 4. Find the numbers.

Given that the three numbers are in the ratio 1:2:3, we can represent the numbers as \( 4x \), \( 8x \), and \( 12x \), where \( x \) is a common multiplier.

The Highest Common Factor (HCF) of these numbers is given as 4. Since \( x \) is multiplied by 4 in all the numbers, the HCF of the expressions \( 4x \), \( 8x \), and \( 12x \) is \( 4 \times \text{HCF of } x \). 

This means:

\[
\text{HCF of } x = 1
\]

Thus, \( x = 1 \).

So, the three numbers are:
\[
4x = 4 \times 1 = 4
\]
\[
8x = 8 \times 1 = 8
\]
\[
12x = 12 \times 1 = 12
\]

Therefore, the three numbers are **4, 8, and 12**.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How would the problem change if the HCF was 6 instead of 4?
2. What is the least common multiple (LCM) of the numbers 4, 8, and 12?
3. If the ratio was 2:3:4 instead of 1:2:3, how would that affect the numbers?
4. How do you find the HCF of more than two numbers?
5. What is the sum of the three numbers 4, 8, and 12?

**Tip:** When dealing with ratios, remember that multiplying all terms by the same number preserves the ratio and scales the numbers proportionally.

Learn how to solve the problem of finding three numbers in the ratio 1:2:3 with a Highest Common Factor (HCF) of 4. Understand the concept of ratios, how to determine the actual numbers from a given ratio, and apply the concept of HCF in number theory. Suitable for students in Grades 7-9.

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