The expenses on rice, fish and oil of a family are in the ratio 12 : 17 : 3. The prices of these articles are increased by 20%, 30% and 50% respectively. The total expenses of family on these articles are increased by (S.S.C., 2007)

### Concept & Overall Percentage Change The overall percentage increase of a total is the sum of the absolute increases divided by the original total. $$ \text{Overall \% Increase} = \frac{\text{Total Increase}}{\text{Original Total}} \times 100 $$ ### Step-by-Step Solution * **Set Initial Values**: Let the expenses be $120$, $170$, and $30$ (scaling the ratio $12:17:3$ by $10$ for easy percentage calculations). * **Find Initial Total**: $120 + 170 + 30 = 320$. * **Calculate Individual Increases**: * Rice increase: $20\%$ of $120 = \frac{20}{100} \times 120 = 24$. * Fish increase: $30\%$ of $170 = \frac{30}{100} \times 170 = 51$. * Oil increase: $50\%$ of $30 = \frac{50}{100} \times 30 = 15$. * **Calculate Total Increase**: $24 + 51 + 15 = 90$. * **Calculate Percentage Increase**: $$\text{Percentage Increase} = \frac{90}{320} \times 100 = \frac{9}{32} \times 100 = \frac{900}{32}$$ $$\frac{900}{32} = \frac{225}{8} = 28.125\%$$ * Convert to a mixed fraction: $28 \frac{1}{8}\%$. ### Exam Strategy & Shortcut Use weighted averages. The overall percentage increase is the weighted sum of the individual percentage increases: $$\text{Overall Change} = \frac{(12 \times 20) + (17 \times 30) + (3 \times 50)}{12 + 17 + 3}$$ $$\text{Overall Change} = \frac{240 + 510 + 150}{32} = \frac{900}{32} = \frac{225}{8} = 28 \frac{1}{8}\%$$ ### Common Pitfall Averaging the three percentage values directly ($20 + 30 + 50) / 3$ without taking into account the unequal weights of the initial expenses. ### Final Answer Therefore, the correct answer is **$28 \frac{1}{8}\%$**.