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# Hardest Question I've ever seen!

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A worker's salary was reduced by m%. Find the percent by which the new salary would have to be raised to bring it back to the original amount.

Feb 9, 2018

#1
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Call S  the old  salary

The new salary   is

S ( 1 -  m )      [ where m is some percent  expessed as a decimal ]

So......the percent, P , that the new salary must be raised to equal the old is

S (1 - .m) ( P + 1)  = S      divide both sides by   S (1 - .m)

( P  + 1) =   S / [ S ( 1 - m  )  ]

P + 1  =  1  / ( 1 - m)

P  =  1 /  ( 1  - m )   - 1

For example....suppose a man makes \$100 per day and has his salary reduced by 5%

New salary  =   100 (1 -. 05)  =  \$95

So....the percent that this must be raised to return to the original salary is

(P + 1)  =  1  /  (1 - .05)

P + 1   ≈  1.0526

P   ≈  .0526  ≈  5.26%   Feb 9, 2018
edited by CPhill  Feb 9, 2018
#2
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If the original salary was =S, and:

If the lower salary was = P, then we have:

S x [1 - m%] = P

S x [1 - m/100] = P divide both sides by [1 - m/100]

S = P / [ 1 - m/100], or:

S = P x 1/[1 - m/100]. Therefore:

[1 - m%] % =[1 - m/100] -  This is the % by which the current salary P must be divided or multiplied by its reciprocal in order to get the original salary S.

Feb 9, 2018