Ken had 1 1/2 times as many bookmarks as Lionel. Mike had 3/4 as many bookmarks as Ken. Ken and Lionel gave Mike a number of bookmarks in the ratio 3:1. As a result, Mike nad twice as many bookmarks as before, and Lionel had 32 bookmarks more than Ken. How many bookmarks did Ken give to Mike?

Answer 1

"Ken had 1 1/2 times as many bookmarks as Lionel" :
Number for Lionel ---- x
number for Ken = (3/2)x
"Mike had 3/4 as many bookmarks as Ken":
number for Mike = (3/4)(3/2)x = (9/8)x

"Ken and Lionel gave Mike a number of bookmarks in the ratio 3:1":
number Ken gave to Mike ---- 3y
number Lionel gave to Mike --- y , (note ratio of 3y : y = 3:1 )
Mike now has (9/8)x + 4y
Ken now has (3/2)x - 3y
Lionel now has x - y

"Mike nad twice as many bookmarks as before"
-----> (9/8)x + 4y = 2((9/8)x
times 8
9x + 32y = 18x or -9x + 32y = 0
9x - 32y = 0 **
"Lionel had 32 bookmarks more than Ken"
----> x-y - ( (3/2)x - 3y ) = 32
x - y - (3/2)x + 3y = 32
2x - 2y - 3x + 6y = 64
-x + 4y = 64 -----> -9x + 36y = 576 ***
add ** and ***
4y = 576
y = 144
in 9x - 32y = 0
9x - 4608 = 0
x = 512

Leaving it up to you to verify and state the conclusion,
(btw, it works!)

Answer 2

WRONG!!!

Answer 3

Number for Lionel ---- x ---------> 512
number for Ken = (3/2)x ---------> 768
"Mike had 3/4 as many bookmarks as Ken":
number for Mike = (3/4)(3/2)x = (9/8)x -------> 576

after exchange:
Mike now has (9/8)x + 4y -------> 1152
Ken now has (3/2)x - 3y ------> 336
Lionel now has x - y -----> 368

Does Mike have twice as many bookmarks as before ?
Is 1152 twice 576 ? YES!
Does Lionel have 32 bookmarks more than Ken ??
Well, Lionel now has 368 and Ken has 336, isn't that 32 more ????? YES