Set initial values

Assume initial income is $100$. Since $75\%$ is spent, initial expenditure is $75$ and initial savings = $100 - 75 = 25$.

$S_{initial} = 25$

Calculate new values

Income increases by $20\%$, so new income = $100 + 20 = 120$. Expenditure increases by $10\%$, so new expenditure = $75 + (0.10 \times 75) = 75 + 7.5 = 82.5$.

$E_{new} = 82.5$

Calculate new savings

Subtract new expenditure from new income: $120 - 82.5 = 37.5$.

$S_{new} = 37.5$

Determine percentage increase

The change in savings is $37.5 - 25 = 12.5$. The percentage increase is calculated as $\frac{12.5}{25} \times 100$.

$\frac{12.5}{25} \times 100 = 50\%$