SOLUTION: Sue is five years youger than Candice.The product of thier ages two years from now will be eighteen more than twice the product of their ages a year ago.How old are Sue and Candice

Question 584734: Sue is five years youger than Candice.The product of thier ages two years from now will be eighteen more than twice the product of their ages a year ago.How old are Sue and Candice now?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let s represent sue's age now.
let c represent candice's age now.
your answer is:
s equals 4 and c equals 9.
2 years from now s will be equal to 6 and c will be equal to 11.
1 year ago s was equal to 3 and c was equal to 8.
the problems states that the product of their ages 2 years from now will be 18 more than twice the product of their ages a year ago.
using the solutions found, this equation becomes:
6 * 11 = 18 + 2 * 3 * 8 which simplifies to:
66 = 18 + 48
simplify this to get:
66 = 66 which confirms the numbers are good.
how did we get it?
today sue is 5 years younger than candice.
this translates to:
s = c - 5
2 years from now the product of sue's age and candice's age will be:
(s + 2) * (c - 5 + 2) which becomes:
(s + 2) * (c - 3)
1 year ago, the product of sue's age and candice's age was:
(s - 1) * (c - 5 - 1) which became:
(s - 1) * (c - 6)
the equations we will be working with are:
(s + 2) * (c - 3) which is the product of their ages 2 years from now.
(s - 1) * (c - 6) which is the product of their ages 1 year ago.
the formula we will be using is:
product of their ages 2 years from now is equal to 18 plus [ 2 times the product of their ages 1 year aqo ].
this becomes:
(s+2) * (c-3) = 18 + [ 2 * (s-1) * (c-6) ]
you need to multiply these out to get:
sc - 3s + 2c - 6 = 18 + 2 * (sc - 6s - c + 6)
simplify this to get:
sc - 3s + 2c - 6 = 18 + 2sc - 12s - 2c + 12
add 12s and 2c to both sides of this equation to get:
sc - 3s + 12s + 2c + 2c - 6 = 18 + 2sc + 12
simplify this by combining like terms to get:
sc + 9s + 4c - 6 = 30 + 2sc
subtract 2sc from both sides of this equation to get:
-sc + 9s + 4c - 6 = 30
add 6 to both sides of this equation to get:
-sc + 9s + 4c = 36
since s = c - 5, substitute for s in this equation to get:
-((c-5)*c) + 9*(c-5) + 4c = 36
simplify to get:
-c^2 + 5c + 9c - 45 + 4c = 36
combine like terms to get:
-c^2 + 18c - 45 = 36
add 45 to both sides of this equation to get:
-c^2 + 18c = 81
add c^2 and subtract 18c from both sides of this equation to get:
0 = c^2 - 18c + 81 which is the same as:
c^2 - 18c + 81 = 0
factor this quadratic equation to get:
(c-9) * (c-9) = 0
this results in c = 9.
since s = c - 5, this results in s = 4
your answer is s = 4 and c = 9 as shown and confirmed way up top at the beginning of this exercise.

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