SOLUTION: Ten years from now Charlotte will be twice as old as Emily will be then. Five years ago Charlotte was 5 times as old as Emily was then. How old are both now?

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Question 595521: Ten years from now Charlotte will be twice as old as Emily will be then. Five years ago Charlotte was 5 times as old as Emily was then. How old are both now?
Answer by math-vortex(648) About Me  (Show Source):
You can put this solution on YOUR website!
Ten years from now Charlotte will be twice as old as Emily will be then. Five years ago Charlotte was 5 times as old as Emily was then. How old are both now?
.
Hi, there!
.
[I] Define the variables.
C = Charlotte's age now
E = Emily's age now
.
[II] Write a system of equations.
In the first sentence, we translate "ten years from now Charlotte will be" into algebra as C+10.
We translate "as old as Emily will be then" into algebra as E+10.
"Charlotte will be twice as old as Emily" so our first equation is
C%2B10=%282%29%28E%2B10%29
.
In the second sentence, we translate "Five years ago Charlotte was" as C-5.
We translate "Emily was then" as E-5.
"Charlotte was five times as old as Emily" so our second equation is
C-5=%285%29%28E-5%29
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[III] Simplify and solve the system of equations.
C%2B10=%282%29%28E%2B10%29
C%2B10=2E%2B20%29
C=2E%2B10%29
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Now that we have isolate the variable C, we can substitute 2E+10 for C in the second equation. Simplify and solve for E.
.
C-5=%285%29%28E-5%29
%282E%2B10%29-5=%285%29%28E-5%29
2E%2B5=5E-25
-3E%2B5=-25
-3E=-30
E=10
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Now we know that Emily is 10 years old. To find Charlotte's age, substitute 10 for E in either equation.
C%2B10=%282%29%28E%2B10%29
C%2B10=%282%29%2810%2B10%29
C%2B10=%282%29%2820%29
C%2B10=40
C=30
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We see that Charlotte is 30 years old. Now go back to the original sentences and make sure that the ages work.
.
Ten years from now Charlotte will be twice as old as Emily will be then.
[30+10] is twice as old as [10+10], or 40 is twice as old as 20. YES!
.
Five years ago Charlotte was 5 times as old as Emily was then.
[30-5] was 5 times as old as [10-5], or 25 is five times 5. YES!
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I hope that this helps! Feel free to email if you have questions.
.
Ms.Figgy
math.in.the.vortex@gmail.com