Question 1194289: The ages of two sisters are 9 and 12 years respectively. In how many years time will the product of their ages be 418 years? Found 2 solutions by Boreal, greenestamps:Answer by Boreal(15235) (Show Source):
You can put this solution on YOUR website! x is the number of years from now.
(9+x)(12+x)=418
108+9x+12x+x^2=418
x^2+21x-310=0
(x+31)(x-10)=0
x=10 years from now, only positive root.
they will be 19 and 22, and that product is 418.
The response from the other tutor shows a perfectly good formal algebraic solution -- which is probably what you were looking for.
But you can get some very good mental exercise by solving the problem using logical reasoning. Here is one way to do that.
(1) Their ages are not far apart. So think of them as "about the same", so that the product of their ages will be close to a perfect square.
(2) Then look for a perfect square that is close to 418. 400 = 20^2 is close. So the two ages 10 years from now will be two whole numbers close to 20 whose product is 418 and whose difference is 3.
(3) The last digit of the product is 8; two numbers with a difference of 3 and close to 20 which multiplied together give last digit 8 are 19 and 22.
Then multiplying 22*19 shows that those are the correct ages 10 years from now, so their current ages are 9 and 12.
