How to work out percentage change with inverse proportionality?

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$$a \propto \frac{1}{b}$$

If $a$ increases by 10%, how would one work out the percentage change of $b$, without knowing the value of either variable?

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Inverse proportion means: $a\cdot b=k$.

Hence: $$1.1a\cdot \frac{1}{1.1}b=k \Rightarrow \frac{1}{1.1}-1=-\frac{1}{11}.$$