To solve the problem where the ratio of two numbers is 8:3 and their sum is 33, we can follow these steps:
1. Define the Ratio Variables:
Let's denote the two numbers as [tex]\(8x\)[/tex] and [tex]\(3x\)[/tex]. Here, [tex]\(x\)[/tex] is a common multiple that maintains the ratio between the two numbers.
2. Set Up the Sum Equation:
According to the problem, the sum of the two numbers equals 33.
[tex]\[
8x + 3x = 33
\][/tex]
3. Simplify the Equation:
Combine the terms on the left side of the equation.
[tex]\[
11x = 33
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation by 11 to find the value of [tex]\(x\)[/tex].
[tex]\[
x = \frac{33}{11} = 3
\][/tex]
5. Calculate the Actual Numbers:
Substitute [tex]\(x = 3\)[/tex] back into the expressions for the numbers.
[tex]\[
\text{First number} = 8x = 8 \times 3 = 24
\][/tex]
[tex]\[
\text{Second number} = 3x = 3 \times 3 = 9
\][/tex]
6. Find the Positive Difference:
Subtract the second number from the first to obtain the positive difference.
[tex]\[
\text{Positive difference} = |24 - 9| = 15
\][/tex]
Therefore, the positive difference between the two numbers is:
[tex]\(\boxed{15}\)[/tex]
