document.write( "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? \n" ); document.write( "

Algebra.Com's Answer #154513 by ichudov(507) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let's say Dorothy age is d, Ricardo's age is r:\r
\n" ); document.write( "\n" ); document.write( " $\"system%28+d=r%2B6%2C+d%2Ar+=+2%2A%28d-6%29%2A%28r-6%29%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "Replace d with r+6:\r
\n" ); document.write( "\n" ); document.write( "(r+6)*r = 2*(r+6-6)*(r-6)\r
\n" ); document.write( "\n" ); document.write( "Give this problem to my universal simplifier:\r
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Solved by pluggable solver: EXPLAIN simplification of an expression

Your Result:

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\n" ); document.write( " YOUR ANSWER\n" ); document.write( "

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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.

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DETAILED EXPLANATION

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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 $\"%28r%2B6%29%2Ar=2%2A%28r%2Bhighlight_green%28+0+%29%29%2A%28r-6%29\"$ .
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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 $\"%28r%2B6%29%2Ar=2%2A%28r%29%2A%28r-6%29\"$ .
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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 $\"%28r%2B6%29%2Ar=2%2Ahighlight_green%28+r+%29%2A%28r-6%29\"$ .
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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 $\"%28r%2B6%29%2Ar-highlight_green%28+2%2Ar%2A%28r-6%29+%29=0\"$ .
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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 $\"highlight_green%28+r%2Ar+%29%2Bhighlight_green%28+r%2A6+%29-2%2Ar%2A%28r-6%29=0\"$ .
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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 $\"highlight_green%28+r%5E2+%29%2Br%2A6-2%2Ar%2A%28r-6%29=0\"$ .
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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 $\"r%5E2%2Br%2A6-highlight_green%28+2%2Ar%2Ar+%29%2Bhighlight_green%28+2%2Ar%2A6+%29=0\"$ .
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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 $\"r%5E2%2Br%2A6-2%2Ar%2Ar%2Bhighlight_green%28+12+%29%2Ar=0\"$ .
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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 $\"r%5E2%2Br%2A6-2%2Ahighlight_green%28+r%5E2+%29%2B12%2Ar=0\"$ .
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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 $\"highlight_green%28+%281-2%29%2Ar%5E2+%29%2Br%2A6%2B12%2Ar=0\"$ .
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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 $\"%281-2%29%2Ar%5E2%2Bhighlight_green%28+%286%2B12%29%2Ar+%29=0\"$ .
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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 $\"%28highlight_green%28+-1+%29%29%2Ar%5E2%2B%286%2B12%29%2Ar=0\"$ .
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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 $\"%28-1%29%2Ar%5E2%2B%28highlight_green%28+18+%29%29%2Ar=0\"$ .
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Look at $\"%28highlight_red%28+-1+%29%29%2Ar%5E2%2B%2818%29%2Ar=0\"$ .
Removed extra sign in front of $\"-1\"$
It becomes $\"%28-highlight_green%28+1+%29%29%2Ar%5E2%2B%2818%29%2Ar=0\"$ .
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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 $\"-highlight_green%28+1+%29%2Ar%5E2%2B%2818%29%2Ar=0\"$ .
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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 $\"-1%2Ar%5E2%2Bhighlight_green%28+18+%29%2Ar=0\"$ .
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Look at $\"-highlight_red%28+1+%29%2Ar%5E2%2B18%2Ar=0\"$ .
Remove extraneous '1' from product $\"highlight_red%28+1+%29\"$
It becomes $\"-r%5E2%2B18%2Ar=0\"$ .
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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: $\"0=0\"$
\n" ); document.write( "This is an equation! Solutions: r=0,r=18.

Universal Simplifier and Solver

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Done!

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\n" ); document.write( "\n" ); document.write( "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. \r
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