Let's solve this problem step by step.
1. We know that [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( R \)[/tex] are successive even natural numbers in ascending order. Therefore, we can represent them as:
[tex]\[
Q = P + 2
\][/tex]
[tex]\[
R = P + 4
\][/tex]
2. According to the problem, five times [tex]\( R \)[/tex] is eight more than seven times [tex]\( P \)[/tex]. We can write this relationship as an equation:
[tex]\[
5R = 7P + 8
\][/tex]
3. Substitute [tex]\( R = P + 4 \)[/tex] into the equation:
[tex]\[
5(P + 4) = 7P + 8
\][/tex]
4. Distribute the 5 on the left-hand side:
[tex]\[
5P + 20 = 7P + 8
\][/tex]
5. Rearrange the equation to isolate [tex]\( P \)[/tex] on one side:
[tex]\[
5P + 20 - 7P = 8
\][/tex]
Simplify further:
[tex]\[
-2P + 20 = 8
\][/tex]
6. Solve for [tex]\( P \)[/tex]:
[tex]\[
-2P = 8 - 20
\][/tex]
[tex]\[
-2P = -12
\][/tex]
[tex]\[
P = 6
\][/tex]
7. Now that we have [tex]\( P = 6 \)[/tex], let's find [tex]\( Q \)[/tex]:
[tex]\[
Q = P + 2
\][/tex]
[tex]\[
Q = 6 + 2
\][/tex]
[tex]\[
Q = 8
\][/tex]
8. For completeness, let's also find [tex]\( R \)[/tex]:
[tex]\[
R = P + 4
\][/tex]
[tex]\[
R = 6 + 4
\][/tex]
[tex]\[
R = 10
\][/tex]
So the successive even natural numbers are [tex]\( 6 \)[/tex], [tex]\( 8 \)[/tex], and [tex]\( 10 \)[/tex]. Therefore, the value of [tex]\( Q \)[/tex] is [tex]\( 8 \)[/tex].
