Math

Three numbers X, Y, and Z are in the ratio 2:7:8. If 12 is subtracted from Y, then three numbers form a geometric sequence (in the order X, Y–12, Z). Find X, Y, and Z.

There are two sets of answers I already found one set i am unsure how to find the other one.

x=10,y=20,z=80

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1. let the 3 numbers be 2x, 7x and 8x
then (7x-12)/(2x) = 8x/(7x-12)

If 2x, (7x-12) and 8x form a GP, then
(7x-12)^2 = 2x(8x)
49x^2 - 168x + 144 = 16x^2
33x^2 - 168x + 144 = 0
11x^2 - 56x + 48 = 0
(x - 4)(11x - 12) = 0
so x = 4 or x = 12/11

if x = 4, the original numbers are 8, 28, and 32
check: subtract 12 from 28 to get new numbers of 8, 16 and 32, which do form a GP

if x = 12/11, then the 3 originals are 24/11, 84/11 and 96/11
take 12 away from 84/11 to get -48/11
check:
-48/11 ÷ 24/11 = -2
96/11 ÷ -48/11 = -2

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Reiny

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