The price of sugar increases by 32%. A family reduces its consumption so that the expenditure of the sugar is up only by 10%. If the total consumption of the sugar before the price rise was 10 kg per month, then the consumption of sugar per month at present (in kg) is

### Concept & Formula The relationship between Expenditure ($E$), Price ($P$), and Consumption ($C$) is defined by a simple multiplicative equation: $$ \text{Expenditure} = \text{Price} \times \text{Consumption} $$ When both Price and Expenditure change by specific percentages, we can find the exact change in Consumption using a ratio-based approach derived from index numbers. ### Step-by-Step Solution 1. **Assume Index Values:** Let the original price of sugar be $100$ units per kg. 2. **Given Initial Data:** * Original Consumption ($C_1$) = $10$ kg * Original Expenditure ($E_1$) = $100 \times 10 = 1000$ units 3. **Calculate New Price and Expenditure:** * New Price ($P_2$) = Increased by $32\%$, so it becomes $132$ units per kg. * New Expenditure ($E_2$) = Allowed to increase by only $10\%$. * $E_2 = 1000 + (10\% \text{ of } 1000) = 1100$ units. 4. **Solve for New Consumption:** * Using $E_2 = P_2 \times C_2$, rearrange to find $C_2$. * $$ C_2 = \frac{E_2}{P_2} = \frac{1100}{132} $$ 5. **Simplify the Fraction:** * Divide numerator and denominator by $11$: $\frac{100}{12}$ * Divide numerator and denominator by $4$: $\frac{25}{3}$ 6. **Convert to Mixed Fraction:** * $$ \frac{25}{3} = 8 \frac{1}{3} \text{ kg} $$ ### Exam Strategy & Shortcut Bypass the total expenditure calculation and use the direct proportional multiplier method based on index $100$. The ratio of new consumption to old consumption is the ratio of the new expenditure index to the new price index: $$ \frac{C_2}{C_1} = \frac{100 + \text{Exp Increase}\%}{100 + \text{Price Increase}\%} $$ Substitute the given percentages: $$ \frac{C_2}{10} = \frac{110}{132} $$ Simplify the right side ($\frac{110}{132} = \frac{5}{6}$): $$ C_2 = 10 \times \left( \frac{5}{6} \right) = \frac{50}{6} = \frac{25}{3} = 8 \frac{1}{3} \text{ kg} $$ This solves the problem in two simple lines of arithmetic. ### Common Pitfall A frequent conceptual mistake is additive percentage logic. A student might incorrectly think: "Price went up $32\%$, expenditure goes up $10\%$, so consumption must drop by the difference, $22\%$." This is mathematically false because Price and Consumption are multiplied together, not added. Always use ratios or multipliers. ### Final Answer **Therefore, the correct answer is $8 \frac{1}{3}$.**