SOLUTION: Dorothy is 6 years older than Ricardo. The product of their present ages is twice what the product of their ages was 6 years ago. How old is Dorothy?

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Question 204731: Dorothy is 6 years older than Ricardo. The product of their present ages is twice what the product of their ages was 6 years ago. How old is Dorothy?
Answer by ichudov(507) About Me  (Show Source):
You can put this solution on YOUR website!
Let's say Dorothy age is d, Ricardo's age is r:
system%28+d=r%2B6%2C+d%2Ar+=+2%2A%28d-6%29%2A%28r-6%29%29+
Replace d with r+6:
(r+6)*r = 2*(r+6-6)*(r-6)
Give this problem to my universal simplifier:
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • This is an equation! Solutions: r=0,r=18.
  • Graphical form: Equation %28r%2B6%29%2Ar=2%2A%28r%2B6-6%29%2A%28r-6%29 was fully solved.
  • Text form: (r+6)*r=2*(r+6-6)*(r-6) simplifies to 0=0
  • Cartoon (animation) form: simplify_cartoon%28+%28r%2B6%29%2Ar=2%2A%28r%2B6-6%29%2A%28r-6%29+%29
    For tutors: simplify_cartoon( (r+6)*r=2*(r+6-6)*(r-6) )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at %28r%2B6%29%2Ar=2%2A%28r%2Bhighlight_red%28+6+%29-highlight_red%28+6+%29%29%2A%28r-6%29.
Added fractions or integers together
It becomes %28r%2B6%29%2Ar=2%2A%28r%2Bhighlight_green%28+0+%29%29%2A%28r-6%29.
Look at %28r%2B6%29%2Ar=2%2A%28r%2Bhighlight_red%28+0+%29%29%2A%28r-6%29.
Remove extraneous zero highlight_red%28+0+%29
It becomes %28r%2B6%29%2Ar=2%2A%28r%29%2A%28r-6%29.
Look at %28r%2B6%29%2Ar=highlight_red%28+2%2A%28highlight_red%28+r+%29%29%2A%28r-6%29+%29.
Remove unneeded parentheses around factor highlight_red%28+r+%29
It becomes %28r%2B6%29%2Ar=2%2Ahighlight_green%28+r+%29%2A%28r-6%29.
Look at %28r%2B6%29%2Ar=highlight_red%28+2%2Ar%2A%28r-6%29+%29.
Moved these terms to the left highlight_green%28+-2%2Ar%2A%28r-6%29+%29
It becomes %28r%2B6%29%2Ar-highlight_green%28+2%2Ar%2A%28r-6%29+%29=0.
Look at highlight_red%28+%28r%2B6%29%2Ar+%29-2%2Ar%2A%28r-6%29=0.
Expanded term r by using associative property on %28r%2B6%29
It becomes highlight_green%28+r%2Ar+%29%2Bhighlight_green%28+r%2A6+%29-2%2Ar%2A%28r-6%29=0.
Look at highlight_red%28+r+%29%2Ahighlight_red%28+r+%29%2Br%2A6-2%2Ar%2A%28r-6%29=0.
Reduce similar several occurrences of highlight_red%28+r+%29 to highlight_green%28+r%5E2+%29
It becomes highlight_green%28+r%5E2+%29%2Br%2A6-2%2Ar%2A%28r-6%29=0.
Look at r%5E2%2Br%2A6-highlight_red%28+2%2Ar%2A%28r-6%29+%29=0.
Expanded term -2%2Ar by using associative property on %28r-6%29
It becomes r%5E2%2Br%2A6-highlight_green%28+2%2Ar%2Ar+%29%2Bhighlight_green%28+2%2Ar%2A6+%29=0.
Look at r%5E2%2Br%2A6-2%2Ar%2Ar%2Bhighlight_red%28+2+%29%2Ar%2Ahighlight_red%28+6+%29=0.
Multiplied numerator integers
It becomes r%5E2%2Br%2A6-2%2Ar%2Ar%2Bhighlight_green%28+12+%29%2Ar=0.
Look at r%5E2%2Br%2A6-2%2Ahighlight_red%28+r+%29%2Ahighlight_red%28+r+%29%2B12%2Ar=0.
Reduce similar several occurrences of highlight_red%28+r+%29 to highlight_green%28+r%5E2+%29
It becomes r%5E2%2Br%2A6-2%2Ahighlight_green%28+r%5E2+%29%2B12%2Ar=0.
Look at highlight_red%28+r%5E2+%29%2Br%2A6-highlight_red%28+2%2Ar%5E2+%29%2B12%2Ar=0.
Eliminated similar terms highlight_red%28+r%5E2+%29,highlight_red%28+-2%2Ar%5E2+%29 replacing them with highlight_green%28+%281-2%29%2Ar%5E2+%29
It becomes highlight_green%28+%281-2%29%2Ar%5E2+%29%2Br%2A6%2B12%2Ar=0.
Look at %281-2%29%2Ar%5E2%2Bhighlight_red%28+r%2A6+%29%2Bhighlight_red%28+12%2Ar+%29=0.
Eliminated similar terms highlight_red%28+r%2A6+%29,highlight_red%28+12%2Ar+%29 replacing them with highlight_green%28+%286%2B12%29%2Ar+%29
It becomes %281-2%29%2Ar%5E2%2Bhighlight_green%28+%286%2B12%29%2Ar+%29=0.
Look at %28highlight_red%28+1+%29-highlight_red%28+2+%29%29%2Ar%5E2%2B%286%2B12%29%2Ar=0.
Added fractions or integers together
It becomes %28highlight_green%28+-1+%29%29%2Ar%5E2%2B%286%2B12%29%2Ar=0.
Look at %28-1%29%2Ar%5E2%2B%28highlight_red%28+6+%29%2Bhighlight_red%28+12+%29%29%2Ar=0.
Added fractions or integers together
It becomes %28-1%29%2Ar%5E2%2B%28highlight_green%28+18+%29%29%2Ar=0.
Look at %28highlight_red%28+-1+%29%29%2Ar%5E2%2B%2818%29%2Ar=0.
Removed extra sign in front of -1
It becomes %28-highlight_green%28+1+%29%29%2Ar%5E2%2B%2818%29%2Ar=0.
Look at highlight_red%28+%28-highlight_red%28+1+%29%29%2Ar%5E2+%29%2B%2818%29%2Ar=0.
Remove unneeded parentheses around factor highlight_red%28+1+%29
It becomes -highlight_green%28+1+%29%2Ar%5E2%2B%2818%29%2Ar=0.
Look at -1%2Ar%5E2%2Bhighlight_red%28+%28highlight_red%28+18+%29%29%2Ar+%29=0.
Remove unneeded parentheses around factor highlight_red%28+18+%29
It becomes -1%2Ar%5E2%2Bhighlight_green%28+18+%29%2Ar=0.
Look at -highlight_red%28+1+%29%2Ar%5E2%2B18%2Ar=0.
Remove extraneous '1' from product highlight_red%28+1+%29
It becomes -r%5E2%2B18%2Ar=0.
Look at highlight_red%28+-r%5E2%2B18%2Ar+%29=0.
Equation highlight_red%28+-r%5E2%2B18%2Ar=0+%29 is a quadratic equation: -r^2+18*r =0, and has solutions 0,18
It becomes highlight_green%28+0+%29=0.
Result: 0=0
This is an equation! Solutions: r=0,r=18.

Universal Simplifier and Solver


Done!


We can dismiss r=0 answer because it is nonsensical, so we have, r=18. Since d=r+6, dorothy's age = 18+6 = 24.