Hi, there--

The Problem
Four years from now a mother will be 5 times the age of her daughter. At present she will be 9 times 
the age of her daughter.

A Solution

First, we choose our variables:

Let m be the current age of the mother.
Let d be the current age  of the daughter.

The age of the mother five years from now is m+4.
The age of the daughter five years from now is d+4.

Now, we write two equations using our variables that describe the relationships between the age of 
the mother and the age of the daughter. 

The first half of the problem states, "Four years from now a mother will be 5 times the age of her 
daughter." In algebra, we can write this relationship as,



The second part of the problem states, "At present she will be 9 times the age of her daughter."
In algebra, we can write,



We now have a system of two equations with two variables. We will solve this using the substitution 
method. (The elimination method would also work, of course.)




Substitute 9d for m in the first equation.




We solve for d. Clear the parentheses using the distributive property.


Isolate the variable d by subtracting 5d from both sides of the equation.


Combine like terms. 9d-5d=4d on the left side and 5d-5d=0 on the right side.


Subtract 5 from both sides.


Simplify.


Divide both sides of the equation by 4 in order to find the value of d.



In the context of this problem, the equation d=4 means that the daughter is currently 4 years old. To 
find the mother's current age, substitute 4 for d in either of the original equations. (I'll use the second.)





The mother is currently 36 years old.

Our last step is to heck our answers in both equations. Often, if you make an error. Your answer will 
work in one equation but not the other. ALWAYS check both equations! (o:

Substitute 4 for d and 36 for m in each equation.





Good! The two sides of the equation are equal.




Good!

You can also check by using the ages you found in the written problem statement:
"Four years from now a mother will be 5 times the age of her daughter."
Four years from now, the mother will be 36+4=40 years old. The Daughter will be 4+4=8 years old. 
Since 40 is five times eight, our numbers work for the first part of the problem.

"At present she will be 9 times the age of her daughter."
At present the mother is thirty-six years old, and the daughter is nine years old. Thirty-six is nine 
times four, so part two is also satisfied.

That's it. You may email me at math.in.the.vortex@gmail.com  if you have questions about the solution. 

Mrs.Figgy