p + q : q + r : r + p = 4 : 5 : 6 and p + q + r = 30, find q =?

Question: p + q : q + r : r + p = 4 : 5 : 6 and p + q + r = 30, find q =?

Solution:
Given that,
p + q : q + r : r + p = 4 : 5 : 6
and p + q + r = 30 ......(1)

Let,
p + q = 4k, q + r = 5k, r + p = 6k

Now,
⇒ p + q + q + r + r + p = 4k + 5k + 6k
⇒ 2(p + q + r) = 15k
⇒ p + q + r = 15k/2
⇒ 15k/2 = 30 [from 1]
⇒ k = 60/15
∴ k = 4

And,
⇒ r + p = 6k
⇒ r + p = 6 × 4
∴ r + p = 24 .....(2)

From (1),
⇒ p + q + r = 30
⇒ q = 30 - (r + p)
⇒ q = 30 - 24
∴ q = 6