Step 1: Understanding the Question:
The problem states that the price of sugar has increased, but the total money spent on sugar (expenditure) must remain constant. We need to find the percentage decrease in sugar consumption to achieve this.
Step 2: Key Formula or Approach:
The relationship between expenditure, price, and consumption is:
\[ \text{Expenditure} = \text{Price} \times \text{Consumption} \]
If the expenditure is constant, then Price is inversely proportional to Consumption.
A shortcut formula for this type of problem is:
\[ \text{Percentage Reduction} = \left( \frac{r}{100 + r} \right) \times 100% \]
where \(r\) is the percentage increase in price.
Step 3: Detailed Explanation:
Let the initial price of sugar be \(P_{1}\) and the initial consumption be \(C_{1}\).
The initial expenditure is \(E_{1} = P_{1} \times C_{1}\).
The price increases by 25%. The new price, \(P_{2}\), is:
\[ P_{2} = P_{1} + 25% \text{ of } P_{1} = P_{1} + 0.25 P_{1} = 1.25 P_{1} \]
Let the new consumption be \(C_{2}\). The new expenditure is \(E_{2} = P_{2} \times C_{2}\).
The problem states that the expenditure remains the same, so \(E_{1} = E_{2}\).
\[ P_{1} \times C_{1} = P_{2} \times C_{2} \]
\[ P_{1} \times C_{1} = (1.25 P_{1}) \times C_{2} \]
Now, we solve for \(C_{2}\) in terms of \(C_{1}\):
\[ C_{2} = \frac{P_{1} \times C_{1}}{1.25 P_{1}} = \frac{C_{1}}{1.25} = \frac{C_{1}}{5/4} = \frac{4}{5} C_{1} = 0.8 C_{1} \]
The reduction in consumption is \(C_{1} - C_{2} = C_{1} - 0.8 C_{1} = 0.2 C_{1}\).
To find the percentage reduction, we use the formula:
\[ \text{Percentage Reduction} = \frac{\text{Reduction in Consumption}}{\text{Initial Consumption}} \times 100 \]
\[ \text{Percentage Reduction} = \frac{0.2 C_{1}}{C_{1}} \times 100 = 0.2 \times 100 = 20% \]
Step 4: Final Answer:
The household must reduce consumption by 20% to keep the expenditure the same.