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Given that $a$ and $b$ are positive integers and that $a+b=24$, what is the value of $ab$ if $2ab + 10a = 3b + 222$?

 Apr 22, 2018
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Given that $a$ and $b$ are positive integers and that $a+b=24$, what is the value of $ab$ if $2ab + 10a = 3b + 222$?

 

a + b  =  24      so    b = 24 - a

 

Sub this into the other equation

 

2a (24 - a)  + 10a  = 3(24 - a) + 222       simpllify

 

48a - 2a^2  + 10a  = 72 - 3a  + 222

 

-2a^2 + 58a   = 294 - 3a     rearrange as

 

2a^2 - 61a + 294  = 0  factor this as

 

(2a - 49)(a - 6)  =  0

 

Only the second factor will produce an integer if set to 0  ...so  a - 6  = 0  ⇒  a = 6

And b  = 24 - 6  = 18

 

So   ab  =  6*18   =  108

 

 

cool cool cool

 Apr 22, 2018

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