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Adam has some brown and green buttons in a box. If he were to remove some brown buttons, the ratio of brown to green buttons would then be 3 : 2. If he were to remove the same number of green buttons, the ratio of brown to green buttons would then be 14 : 1. Find the ratio of brown buttons to green buttons in the box.

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May 17, 2021

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We  have two equations

(G - x ) / B  =   3/2     ⇒     2(G - x)  =  3B  ⇒  2G - 2x  =  3B  ⇒  (28/3)G - (28/3)x =  14B   (1)

G  /( B - x)  =  14   ⇒  G  = 14(B - x)  ⇒  G = 14B  - 14x  ⇒  14B =  G  + 14x   (2)

Equating (1)  and (2)

(28/3)G  - (28/3)x   =  G + 14x

(25/3)G  =  (70/3)x

25G  = 70x

LCM  25, 70    =  350

25G  =350    and  70x  = 350

We  have  an equality  when   G = 14   and  x  =  5

So

2G - 2x =  3B

28 - 10  =  3B

18  = 3B

6 = B

Proof

(14-5) / 6  =  9/6   =  3/2

and

14  / ( 6 - 5)  =     14  / 1

Ratio of B /G   =   14 / 6   =   7 /  3

May 17, 2021
edited by CPhill  May 17, 2021