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Question 714329: What two numbers are twice as far from 40 as they are from 70
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! THE FIFTH GRADER WAY:
That number could be in between 40 and 70,
or could be larger than 70.
(It cannot be less than 40 because that would put it close to 40 than to 70).
If it's in between, it should be  of the way from 40 to 70,
so you have gone 2 thirds of the way, but you have 1 third of the way to go.
Since that way is  ,  of that way is  , and
of that is  .
The number between 40 and 70, that is 10 away from 70, and
away from  is
 .
If the number is larger than 70, and
going from 40 to that number is twice as far as
going from 70 to that number,
then 70 is half way there, and the distance from 40 to 70,
 ,
is the same as from  further to the number,
so the number is  past .
It is 
TWO WAYS THAT LOOK MORE LIKE ALGEBRA:
A number  is
 from and
 from .
We know that
At this point,
we can just square both sides of the equal sign and worry about eliminating any possible extraneous solution later.
Or we could examine what those absolute values could be and work for each case.
Squaring both sides we get
 -->  -->  -->  --> 
Dividing both sides by 3 we get
If we are good at factoring, we factor to get
 with solutions
 and 
If we cannot factor, we apply the quadratic formula:
 --> 
with solutions
 -->  --> and
 -->  --> .
Examining case by case the absolute values:
For  ,
 so  and
 so 
transforms into
 -->  -->  -->  -->
For  ,
 so  and
 so 
transforms into
 and multiplying times  both sides of the equal sign we get
as above,
which gives us the solution  as above,
but as we had started with ,
we did not find a solution that fit the case .
For such that  ,
so and
so
transforms into
 -->  -->  -->  -->  -->  -->  --> --> .
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