Question 995661
Each of your starting scales is related to the ideal scale as a line.


Points (x,y) using x for IQ value on one scale, and y for IQ value on the ideal scale.  The ideal scale genius value would be average of 130 and 160, being 145 for genius on the ideal scale.


Your "one scale" and your ideal scale will have these points:
(0,0) and (130,145).
The linear to relate these two points has slope  {{{(145-0)/(130-0)=145/130=29/26}}}.
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So far this gives you an equation {{{y=(29/26)x+b}}}, and you want to use either point to find b.  I suggest trying .... does not matter.  Both ideal and "one scale" have vertical axis intercept of 0.
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The formula for relating y, ideal IQ to the "one scale" IQ is simply {{{highlight(y=(29/26)x)}}}.


Person A having IQ of 138 on the "one scale" will have an "ideal" IQ of  {{{y=(29/26)(138)=154}}}.


Maybe you can find the relationship for the "different scale".  


Simple familiarity with equations and points for lines is necessary for the best most comfortable solving ability with this problem.  This is much like the very typical F & C temperature scale conversion question.