The price of sugarcane rises by $20\%$. By how much percent must a housewife reduce her consumption so that her expenditure remains the same?

Let the initial price be Rs. $100$ and initial consumption be $100$ units.
$$\text{Initial Expenditure}=100\times 100=10000$$

The price increases by $20\%$, making the new price Rs. $120$. Let the new consumption be $C$.
Since the total expenditure must remain unchanged:

$$120\times C=10000$$

$$C=\frac{10000}{120}=\frac{250}{3}=83\frac{1}{3}\text{ units}$$

Reduction in consumption:

$$\text{Reduction}=100-83\frac{1}{3}=16\frac{2}{3}\%$$

Alternative Shortcut Formula:

$$\text{Percentage Reduction}=\left(\frac{R}{100+R}\times 100\right)\%$$

where $R=20$:

$$\text{Reduction}=\left(\frac{20}{120}\times 100\right)\%=\left(\frac{1}{6}\times 100\right)\%=16\frac{2}{3}\%$$