The percentage, if the new cost was 84% of the original is 40%
Let the cost of the item be x.
When the price increased by y%, the new price is:
![New =x \times (1 + y\%)](https://tex.z-dn.net/?f=New%20%3Dx%20%5Ctimes%20%281%20%2B%20y%5C%25%29)
When the price decreased by the same percentage (i.e. y%), the new price is:
![New =x \times (1 + y\%) \times (1 - y\%)](https://tex.z-dn.net/?f=New%20%3Dx%20%5Ctimes%20%281%20%2B%20y%5C%25%29%20%5Ctimes%20%281%20-%20y%5C%25%29)
The cost is said to be 84% of the original price.
So, we have:
![x \times (1 + y\%) \times (1 - y\%) =84\% \times x](https://tex.z-dn.net/?f=x%20%5Ctimes%20%281%20%2B%20y%5C%25%29%20%5Ctimes%20%281%20-%20y%5C%25%29%20%3D84%5C%25%20%5Ctimes%20x)
Divide both sides by x
![(1 + y\%) \times (1 - y\%) =84\%](https://tex.z-dn.net/?f=%281%20%2B%20y%5C%25%29%20%5Ctimes%20%281%20-%20y%5C%25%29%20%3D84%5C%25)
Apply the difference of two squares
![1 - (y\%)^2 =84\% \\\\](https://tex.z-dn.net/?f=1%20-%20%28y%5C%25%29%5E2%20%3D84%5C%25%20%5C%5C%5C%5C)
Subtract 1 from both sides of the equation
![- (y\%)^2 =-16\%](https://tex.z-dn.net/?f=-%20%28y%5C%25%29%5E2%20%3D-16%5C%25)
Cancel out the common factors
![(y\%)^2 =16\%](https://tex.z-dn.net/?f=%28y%5C%25%29%5E2%20%3D16%5C%25)
Take the square root of both sides
![y\% =0.4](https://tex.z-dn.net/?f=y%5C%25%20%3D0.4)
Multiply both sides by 100
![y =40](https://tex.z-dn.net/?f=y%20%3D40)
Hence, the percentage is 40%
Read more about percentage change at:
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