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Percentage Unchanged

Age 14 to 16 Short Challenge Level:

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.

You can find more short problems, arranged by curriculum topic, in our short problems collection.