The production cost of an item increased by 20% and the government also started taxing the item 10% on the selling price. The percentage increase in the selling price which will ensure profit remains the same would be ____

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Let the cost of the item be c, then the increased cost is 1.2c.

Let s₁ be the original selling price. Let the new selling price before tax be s₂ then after tax it will be 1.1s₂. The original profit is s₁-c. The new profit is 1.1s₂-1.2c=s₁-c, because the profit is to remain unchanged. Therefore 1.1s₂-s₁=0.2c. If we divide through by s₁ we get 1.1(s₂/s₁)=1+0.2c/s₁. So s₂/s₁=(1+0.2c/s₁)/1.1. If s₂=s₁+rs₁/100 where r is the percentage increase in selling price, then s₂/s₁=1+r/100=(1+0.2c/s₁)/1.1. r/100=(1+0.2c/s₁)/1.1-1=(0.2c/s₁-0.1)/1.1 and r=100(0.2c/s₁-0.1)/1.1=10(2c/s₁-1)/1.1.

This implies a negative value for r, which means a reduction in the selling price, since c/s₁ is 1/(original markup ratio).

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