If the price of a commodity be raised by 40%,by how much percent must a householder reduce his consumption of that commodity, so as not to increase his expenditure?

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Define the original price and consumption
Let the original price of the commodity be \( x \) and the original consumption be \( y \).

### Step 2: Calculate the original expenditure
The original expenditure can be calculated as:
\[ \text{Original Expenditure} = \text{Price} \times \text{Consumption} = x \times y \]

### Step 3: Determine the new price after the increase
The price of the commodity is raised by 40%. Therefore, the new price \( P' \) can be calculated as:
\[ P' = x + 0.4x = 1.4x \]

### Step 4: Set up the equation for new expenditure
Let the new consumption be \( y' \). The new expenditure must remain the same as the original expenditure:
\[ \text{New Expenditure} = P' \times y' = 1.4x \times y' \]
Since the expenditures are equal, we have:
\[ x \times y = 1.4x \times y' \]

### Step 5: Simplify the equation
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ y = 1.4y' \]

### Step 6: Solve for new consumption \( y' \)
Rearranging the equation gives us:
\[ y' = \frac{y}{1.4} \]

### Step 7: Calculate the percentage reduction in consumption
To find the percentage reduction in consumption, we first find the reduction in consumption:
\[ \text{Reduction} = y - y' = y - \frac{y}{1.4} = y \left(1 - \frac{1}{1.4}\right) \]
Calculating \( 1 - \frac{1}{1.4} \):
\[ 1 - \frac{1}{1.4} = 1 - \frac{5}{7} = \frac{2}{7} \]
Thus, the reduction in consumption is:
\[ \text{Reduction} = y \times \frac{2}{7} \]

### Step 8: Calculate the percentage reduction
The percentage reduction in consumption is given by:
\[ \text{Percentage Reduction} = \left(\frac{\text{Reduction}}{y}\right) \times 100 = \left(\frac{y \times \frac{2}{7}}{y}\right) \times 100 = \frac{2}{7} \times 100 \]
Calculating \( \frac{2}{7} \times 100 \):
\[ \frac{2 \times 100}{7} = \frac{200}{7} \approx 28.57\% \]

### Final Answer
The householder must reduce his consumption by approximately \( 28.57\% \).

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