Question 656060: One number is 1/5 of another number. Together the two numbers equal 12. What are the numbers?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! You can easily find the answer the fifth-grader way without even touching a pen or pencil. I would do that first (no need to tell anyone), and then I would try to solve it the algebra way. Knowing the result, I would be able to catch and correct any mistakes in my algebra work. Finally I would only show the work the teacher expected, demonstrating how skillfully I can use algebra to solve the problem the expected way.
THE FIFTH GRADER WAY:
We assume that what is meant by "number" is a positive integer, like 1, 2, 3, 4, etc.
The smaller number is  of the larger number, so the larger number must be divisible by  , like or  .
The larger number cannot be 15 or larger, because then the two numbers would add up to more that  .
If the larger number was , of the larger number would be  ; the two numbers would be and , and added together they would equal  .
If the larger number was , of the larger number would be  ; the two numbers would be and , and added together they would equal  .
The two numbers are and .
THE ALGEBRA SOLUTION
First you have to name your one variable or your two variables.
You could say (or just think) "let the larger number be  ."
You could also say (or just think) "let the smaller number be  ."
Some teachers expect students to write those "let ..." statements. Otherwise, you just think the words I write as your own explanation, but only write the needed equations.
We know that  or  .
(If you are expected to use only one variable, you can use just X for the larger number, and call the other number  ).
Alternatively, you could call the numbers and  ).
We also know that  or  .
Your preference and teacher expectations will determine exactly how to name your variable (or variables) and how to solve the problem. I'll show one of many ways.
 -->  (multiplying both sides of the equal sign time )
-->  -->  (performing operations)
-->  (collecting like terms)
-->  -->  (dividing both sides by 6)
The other number is of ,
so it is 
NOTE:
If you are studying systems of equations (in more advanced algebra), you may be expected to use two variables and write a system of two linear equations with two variables, like
 or 
You could easily solve that system by substitution, going through steps like the ones shown above.
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