Given that the ratio of two numbers is $2:3$ and their highest common factor (HCF) is 15, we can represent the numbers in terms of their ratio as follows:

Let the two numbers be:

• $2x$ and $3x$, where $x$ is a common factor.

Since the HCF of the two numbers is given as 15, and $2$ and $3$ are co-prime (they have no common factor other than 1), $x$ must be equal to the HCF. Therefore, $x = 15$.

Now, substitute $x = 15$ into the expressions for the two numbers:

• First number: $2x = 2 \times 15 = 30$
• Second number: $3x = 3 \times 15 = 45$

Thus, the two numbers are 30 and 45.

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Would you like more details or explanations on any part of this process?

Here are 5 related questions for further exploration:

1. How do you find the HCF of two numbers?
2. What is the difference between the HCF and the LCM?
3. Can two numbers with a ratio of 4:5 have an HCF of 12? Why or why not?
4. How would the solution change if the ratio were 5:7 and the HCF was 10?
5. What is the significance of co-prime numbers in determining HCF?

Tip: The HCF of two numbers is always a factor of both numbers, while their least common multiple (LCM) is a multiple of both numbers.