A man saves $20\%$ of his income. If his expenses are increased by $35\%$, by what percent must his income be raised so that he can save $10\%$ of his new income?

Let the initial income be $100$.
* Initial Savings = $20\%\text{ of }100=20$
* Initial Expenses = $100-20=80$

The expenses increase by $35\%$:

$$\text{New Expenses}=80\times \left(1+\frac{35}{100}\right)=80\times 1.35=108$$

Let the new income be $I$. He wants to save $10\%$ of his new income, which means his expenses will be $90\%$ of his new income:

$$90\%\text{ of }I=108$$

$$\frac{90}{100}\times I=108$$

$$I=\frac{108\times 100}{90}=120$$

The income must increase from $100$ to $120$.

$$\text{Percentage increase in income}=\frac{120-100}{100}\times 100=20\%$$