Question 1158871
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Method 1:


Find the prime factorization of 174
First note that it is an even number, so 
174/2 = 87
174 = 2*87


Then let's factor 87. This is a multiple of 3 because the digits add to 8+7 = 15 which is a multiple of 3
87/3 = 29
87 = 3*29


Therefore,
174 = 2*87
174 = 2*3*29


The mother is 35 and 29 is one of the factors. There is no way to have the son be this old as we're rewinding into the past. The mother can only attain this age. This must mean that 35-29 = 6 years is the answer. 


Six years ago, her son was 12-6 = 6 years old


The product of their ages six years ago was 
29*6 = 174
which confirms our answer


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Method 2:


x = number of years into the past
35-x = woman's age x years ago
12-x = son's age x years ago
(35-x)(12-x) = product of their ages x years ago
(35-x)(12-x) = 35(12-x)-x(12-x) ... distributive property
(35-x)(12-x) = 420-35x-12x+x^2 ... distributive property again
(35-x)(12-x) = 420-47x+x^2
(35-x)(12-x) = x^2-47x+420
note: you can use the FOIL rule or the box method to expand out (35-x)(12-x)


Set this equal to the desired product (174) and get everything to one side
x^2-47x+420 = 174
x^2-47x+420-174 = 0
x^2-47x+246 = 0


Use the quadratic formula to solve for x
{{{x = (-b+sqrt(b^2-4ac))/(2a)}}} or {{{x = (-b-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-(-47)+sqrt((-47)^2-4(1)(246)))/(2(1))}}} or {{{x = (-(-47)-sqrt((-47)^2-4(1)(246)))/(2(1))}}}


{{{x = (47+sqrt(1225))/(2)}}} or {{{x = (47-sqrt(1225))/(2)}}}


{{{x = (47+35)/(2)}}} or {{{x = (47-35)/(2)}}}


{{{x = (82)/(2)}}} or {{{x = (12)/(2)}}}


{{{x = 41}}} or {{{x = 6}}}


Since the woman is 35 years old, it is not possible to go back 41 years into the past before she was born. A similar situation happens with the son as well. So we ignore x = 41 as a solution. The only practical solution is x = 6.


This is the same result as we got with method 1.

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