The price of a product is decreased by \(x\%\) and then increased by \(x\%\). The final price becomes 10% less than the original. What is the value of \(x\) (approximately)?
31.6
Let the original price be 100. Decreasing by \(x\%\) and then increasing by \(x\%\) multiplies the price by \(\left(1-\frac{x}{100}\right)\left(1+\frac{x}{100}\right)\).
Using \((1-a)(1+a)=1-a^2\), the final price \(= 100\left(1-\frac{x^2}{10000}\right)\).
The final price is 10% less than the original, i.e. equal to 90. So \(100\left(1-\frac{x^2}{10000}\right)=90\).
Thus \(1-\frac{x^2}{10000}=0.9\), giving \(\frac{x^2}{10000}=0.1\) and \(x^2=1000\).
Therefore \(x=\sqrt{1000}\approx 31.6\).
Hence, the value of x is approximately 31.6.
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