To solve the problem, we need to determine the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] that satisfy the conditions given:
1. [tex]\(\frac{p}{q} = \frac{5}{2}\)[/tex]
2. [tex]\(2p + q = 108\)[/tex]
Let's proceed step-by-step:
### Step 1: Express [tex]\( q \)[/tex] in terms of [tex]\( p \)[/tex]
From the first condition:
[tex]\[
\frac{p}{q} = \frac{5}{2}
\][/tex]
This can be rearranged to:
[tex]\[
p = \frac{5}{2} q
\][/tex]
### Step 2: Substitute [tex]\( q \)[/tex] in the second equation
We substitute the expression for [tex]\( p \)[/tex] into the second equation:
[tex]\[
2p + q = 108
\][/tex]
Substitute [tex]\( p = \frac{5}{2} q \)[/tex]:
[tex]\[
2 \left( \frac{5}{2} q \right) + q = 108
\][/tex]
This simplifies to:
[tex]\[
5q + q = 108
\][/tex]
Combining like terms:
[tex]\[
6q = 108
\][/tex]
Solving for [tex]\( q \)[/tex], we get:
[tex]\[
q = \frac{108}{6} = 18
\][/tex]
### Step 3: Find the value of [tex]\( p \)[/tex]
Now that we have [tex]\( q = 18 \)[/tex], we use the expression for [tex]\( p \)[/tex] again:
[tex]\[
p = \frac{5}{2} q = \frac{5}{2} \times 18 = 45
\][/tex]
### Step 4: Calculate [tex]\( p - q \)[/tex]
Finally, we calculate [tex]\( p - q \)[/tex]:
[tex]\[
p - q = 45 - 18 = 27
\][/tex]
Therefore, the value of [tex]\( p - q \)[/tex] is [tex]\( \boxed{27} \)[/tex].
