Two positive integers are in the proportion 3:7. If their HCF is 8, determine the integers.

The correct answer is 24, 56.

Step-by-Step Explanation:

Step 1: Understand the relationship between ratio and HCF
When two numbers are given in a simplified ratio, say 3:7, it means that any common factors between them have been divided out. The greatest factor that can be divided out from both numbers is their Highest Common Factor (HCF).

Since the numbers 3 and 7 are co-prime (they share no common factors other than 1), the two original integers can be represented by multiplying each part of the ratio by their HCF, denoted as:

$x$

Therefore, we can write the two numbers as:

$\text{First number}=3\times x$

$\text{Second number}=7\times x$

Step 2: Use the given HCF to find the common multiplier
The problem states that the HCF of the two integers is 8. Therefore:

$x=8$

Step 3: Calculate the exact values of the integers
Substitute the value of $x=8$ into the expressions for the two integers:

For the first integer:

$\text{First number}=3\times 8=24$

For the second integer:

$\text{Second number}=7\times 8=56$

Thus, the two positive integers are 24 and 56.