P, Q and R together have 180 candies among them. P gives Q and R each as many candies as they already have. After this, R gives Q half as many candies as Q already has, and R also gives P twice as many candies as P already has. Now each of them has the same number of candies with them. What is the ratio of the respective number of candies P and R had initially?

Correct option is D

Given:
P, Q and R together have 180 candies
P gives Q and R each as many candies as they already have.
R gives Q half as many candies as Q already has.
R also gives twice as many candies as P already has.
At last each of them has the same number of candies with them.
Solution:
Let the number of candies P, Q and R have be 'p', 'q' and 'r' respectively.
According to the question,
P gives Q and R each as many candies as they already have.
=> Candies left with Q = q + q = 2q
=> Candies left with R = r + r = 2r
=> Candies left with P = p - q - r
R gives Q half as many candies as Q already has
=> Candies left with Q = 2q + q = 3q
=> Candies left with R = 2r - q
R also gives twice as many candies as P already has
=> Candies left with P = (p - q - r) + 2(p - q - r) = 3(p - q - r)
=> Candies left with R = (2r - q) - 2(p - q - r) = 4r + q - 2p
At last each of them has the same number of candies with them
Number of candies left with P = Number of candies left with Q
=> 3(p - q - r) = 3q
=> 3(p - r) = 6q
=> p - r = 2q ....(i)
Number of candies left with R = Number of candies left with Q
=> 4r + q - 2p = 3q
=> 4r - 2p = 2q ....(ii)
Equating equation (i) and (ii)
=> p - r = 4r - 2p
=> 3p = 5r
=> $$\frac pr = \frac 53$$​= 5 : 3

∴ The ratio of the respective number of candies P and R had initially was 5 : 3
Alternate Solution:
Let the number of candies P, Q and R have after last sharing be 60, 60 and 60.
Before last sharing,
=> Candies with P, Q and R was = 20 : 40 : 120
Initial ratio
=> Candies with P, Q and R was = 100 : 20 : 60
=> Ratio of candies with P and R = $$\frac{100}{60}$$​ = 5 : 3
∴ The ratio of the respective number of candies P and R had initially was 5 : 3