Question

Using the properties of proportion,solve:
( 6a^2- ab ):(2ab-b^b)=6:1. Find a:b

Answer 1

first we can convert this into..

(6a^2 - ab) / (2ab - b^2) = 6/1 since r.h.s and l.h.s are ratios

ni we can cros multiply the expression

6a^2 -ab = 12ab - 6b^2

then we can divide the both sides from b^2

6 (a^2/b^2) - (a/b) = 12(a/b) - 6

now lets subtitute
t = (a/b)

then we can get the equation

6t^2 - 13t + 6 =0
6t^2 -4t -9t + 6 =0
2t( 3t - 2) -3(3t -2) =0
(3t -2)(2t -3) =0

sooo
t = 2/3 or t= 3/2

which means
a/b = 2/3 or a/b = 3/2

a:b = 2:3 or a:b = 3:2


both answers satisfy the 1st expression

hope this will help you :)

Answer 2

first we can convert this into..

$(6a^(2) - ab) ()/() (2ab - b^(2)) =( 6)/(1)$ since r.h.s and l.h.s are ratios

then we can cross multiply the expression

$6a^(2) -ab = 12ab - 6b^(2)$

then we can divide the both sides from b^(2)

$6 ((a^(2))/(b^(2))) - ((a)/(b)) = 12((a)/(b)) - 6$

now lets substitute

$t = (a/b)$

then we can get the equation

$6t^(2) - 13t + 6 =0$

$6t^(2) -4t -9t + 6 =0$

$2t( 3t - 2) -3(3t -2) =0$

$(3t -2)(2t -3) =0$

so

$t=2/3$

which means

$(a)/(b) =( 2)/(3)$

$a:b = 2:3$

$thus,$

$a;b = 2:3$