Answer: 9.09090...% 

Using scale factors

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 Area is unchanged so ?% of 110% = 100%

$$\begin{align} \tfrac{?}{100}\times110 &= 100\\

 ?\times110&=10000\\

?\times11&=1000\\

? &= \tfrac{1000}{11} \text{ or } 90.90909...\end{align}$$ So the width is decreased by $100-90.90909...=9.090909...\text{ or }\frac{100}{11}=9\frac1{11}$

Using algebra

The area of the original rectangle was $wb$

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The area of the altered rectangle is $pw \times 1.1b$

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Since the two areas are equal,

$1.1 pwb = wb$

so

$1.1 p = 1$

so $p = \frac{1}{1.1} = \frac{10}{11}$

So the width is $\frac{10}{11}$ of its original value, so it has been decreased by $\frac1{11}$, which as a percentage is $9$ and $\frac1{11}\%$, or $9.09\%$ to 2.d.p.