Answer: 9.09090...% 


Using scale factors
 
 
 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 $wb$

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

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.