let

x = the number of seagulls

y = the number of ducks

there are 108 birds in the a flock

x + y = 108

the ratio of seagulls to ducks is 1:2

x:y = 1:2

y = 2x

first lets find the number of seagulls and ducks by solving the system

y = 2x

x + y = 108

x = 36 seagulls

y = 72 ducks

let 'k' be the number of seagulls to join the flock to change the ratio to 17:24 that is

(36 + k):72 = 17:24

36 + k = (72*17)/24

k = 15

indeed if 15 seagulls joined the flock the ratio will be

(36 + 15) : 72 = 51 : 72 = 17 : 24

15 seagulls must join the flock.